chapters 1 through 5 of Esther, American Standard Version. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information and to find out how you can volunteer, please visit LibriVox.org recording by Sam Stinson chapter 1. Now it came to pass in the days of Ahasuerus. This is Ahasuerus who reigned from India even unto Ethiopia over a hundred and seven and twenty provinces that in those days when the king Ahasuerus sat on the throne of his kingdom which was in Shushan, the palace. In the third year of his reign he made a feast unto all his princes and his servants, the power of Persia and media, the nobles and princes of the provinces being before him. When he showed the riches of his glorious kingdom and the honor of his excellent majesty many days, even a hundred and four score days. And when these days were fulfilled the king made a feast unto all the people that were present in Shushan, the palace, both great and small, seven days in the court of the garden of the king's palace. There were hangings of white cloth, of green and of blue, fastened with cords of fine linen and purple to silver rings and pillars of marble. The couches were of gold and silver upon a pavement of red and white and yellow and black marble and they gave them drink and vessels of gold, the vessels being diverse one from another. And royal wine in abundance according to the bounty of the king and the drinking was according to the law. None could compel for so the king had appointed to all the officers of his house that they should do according to every man's pleasure. Also Vashti, the queen, made a feast for the women in the royal house which belonged to King Ahasuerus. On the seventh day when the heart of the king was married with wine, he commanded Mahuman, Bistha, Harbona, Bigtha, and Abaghtha, Zathar, and Karkis, the seven chamberlands that ministered in the presence of Ahasuerus the king, to bring Vashti the queen before the king, with the crown royal, to show the peoples and the princes her beauty, for she was fair to look on. But the queen Vashti refused to come at the king's commandment by the chamberlands. Therefore was the king very roth and his anger burned in him. Then the king said to the wise men who knew the times, for so was the king's manner toward all that knew law and judgment. And the next unto him were Karshanah, Shethar, Ad-Mathah, Tarshish, Me-res, Mar-senah, and Me-mu-ken, the seven princes of Persia and Media, who saw the king's face, and sat first in the kingdom. What shall we do unto the queen Vashti according to law, because she hath not done the bidding of the king Ahasuerus by the chamberlands? And Me-mu-ken answered before the king and the princes. Vashti the queen hath not done wrong to the king only, but also to all the princes and to all the peoples that are in all the provinces of the king Ahasuerus. For this deed of the queen will come abroad unto all women, to make their husbands contemptible in their eyes, when it shall be reported. The king Ahasuerus commanded Vashti the queen to be brought in before him, but she came not, and this day will the princesses of Persia and Media who have heard of the deed of the queen say the like unto all the king's princes. So will there arise much contempt and wrath. If it please the king, let there go forth a royal commandment from him, and let it be written among the laws of the Persians and the Medes, that it be not altered, that Vashti come no more before king Ahasuerus, and let the king give her royal estate unto another that is better than she. And when the king's decree, which he shall make, shall be published throughout all his kingdom, for it is great. All the wives will give to their husbands honor, both too great and small. And the saying pleased the king and the princes, and the king did according to the word of Me-mu-ken, for he sent letters into all the king's provinces, into every province according to the writing thereof, and to every people after their language, that every man should bear rule in his own house, and should speak according to the language of his people. After these things, when the wrath of king Ahasuerus was pacified, he remembered Vashti and what she had done and what was decreed against her, then said the king's servants that ministered unto him. Let there be fair young virgins sought for the king, and let the king appoint officers in all the provinces of his kingdom, that they may gather together all the fair young virgins unto Shushan, the palace, to the house of the women, unto the custody of Hegai, the king's chamberlain, keeper of the women, and let their things for purification be given them, and let the maiden that pleaseth the king be queen instead of Vashti, and the thing pleaseth the king, and he did so.  And he brought up Hadassah, that is Esther, his uncle's daughter, for she had neither father nor mother, and the maiden was fair and beautiful, and when her father and mother were dead Mordecai took her for his own daughter. So it came to pass when the king's commandment and his decree was heard, and when many maidens were gathered together unto Shushan, the palace, to the custody of Hegai, that Esther was taken into the king's house, to the custody of Hegai, keeper of the women, and the maiden pleased him, and she obtained kindness of him, and he speedily gave her things for purification with her portions, and the seven maidens who were meat to be given her out of the king's house, and he removed her and her maidens to the best place of the house of the women. Esther had not made known her people nor her kindred, for Mordecai had charged her that she should not make it known, and Mordecai walked every day before the court of the women's house, to know how Esther did, and what would become of her. Now, when the turn of every maiden was come to go in to King Ahasuerus, after that it had been done to her according to the law for the women twelve months, for so were the days of their purifications accomplished, to wit, six months with oil of myrrh, and six months with sweet odors, and wit the things for the purifying of the women, then in this wise came the maiden unto the king, whatsoever she desired was given her to go with her out of the house of the women unto the king's house. In the evening she went, and on the morrow she returned into the second house of the women, to the custody of Shaashgaz, the king's chamberlain who kept the concubines. She came in unto the king no more, except the king delighted in her, and she were called by name. Now, when the turn of Esther, the daughter of Abi Ha'il, the uncle of Mordecai, who had taken her for his daughter, was come to go in unto the king, she required nothing but what Haggai the king's chamberlain, the keeper of the women, appointed, and Esther obtained favor in the sight of all them that looked upon her. So Esther was taken unto King Ahasuerus into his house, royal, in the tenth month, which is the month Tebeth, in the seventh year of his reign. And the king loved Esther above all the women, and she obtained favor and kindness in his sight, more than all the virgins, so that he set the royal crown upon her head, and made her queen instead of Vashti. Then the king made a great feast unto all his princes, and his servants, even Esther's feast, and he made a release to the provinces, and gave gifts according to the bounty of the king. And when the virgins were gathered together the second time, then Mordecai was sitting in the king's gate. Esther had not yet made known her kindred, nor her people, as Mordecai had charged her, for Esther did the commandment of Mordecai like, as when she was brought up with him. In those days, while Mordecai was sitting in the king's gate, two of the king's chamberlands, big, thin, and tarish, of those that kept the threshold, were wroth, and sought to lay hands on the king, ahasaris. And the thing became known to Mordecai, who showed it unto Esther, the queen, and Esther told the king thereof in Mordecai's name. And when inquisition was made of the matter, and it was found to be so, they were both hanged on a tree, and it was written in the book of the Chronicles before the king. Chapter 3 After these things did king Ahasaris promote Haman, the son of Ham Medatha, the Agagite, and advanced him and set his seat above all the princes that were with him. And all the king's servants that were in the king's gate bowed down and did reverence to Haman, for the king had so commanded concerning him. But Mordecai bowed not down, nor did him reverence. Then the king's servants that were in the king's gate set unto Mordecai. Why transgresses thou the king's commandment? Now it came to pass when they spake daily unto him, and he hearkened not unto them, that they told Haman to see whether Mordecai's matters would stand, for he had told them that he was a Jew. And when Haman saw that Mordecai bowed not down, nor did him reverence, then was Haman full of wrath. But he thought scorn to lay hands on Mordecai alone, for they had made known to him the people of Mordecai, wherefore Haman sought to destroy all the Jews that were throughout the whole kingdom of Ahasaris, even the people of Mordecai. In the first month, which is the month Nisan in the twelfth year of King Ahasaris, they cast Purr that is the lot before Haman, from day to day, and from month to month, to the twelfth month, which is the month Adar. And Haman said unto King Ahasaris, There is a certain people, scattered abroad and dispersed among the peoples, in all the provinces of thy kingdom, and their laws are diverse from those of every people. Neither keep they the king's laws. Therefore it is not for the king's profit to suffer them. If it pleased the king, let it be written that they be destroyed. And I will pay ten thousand talents of silver into the hands of those that have the charge of the king's business, to bring it into the king's treasuries. And the king took his ring from his hand, and gave it unto Haman, the son of Ham Medatha, the Agagite, the Jews enemy. And the king said unto Haman, The silver is given to thee, the people also, to do with them as it seemeth good to thee. Then were the king's scribes called in, the first month, on the thirteenth day thereof. And there was written according to all that Haman commanded unto the king's satraps, and to the governors that were over every province, and to the princes of every people, to every province according to the writing thereof, and to every people after their language. And the name of King Ahasaris was it written, and it was sealed with the king's ring. And letters were sent by posts into all the king's provinces, to destroy, to slay, and to cause to perish all Jews, both young and old, little children and women, and one day even upon the thirteenth day of the twelfth month, which is the month Adar, and to take the spoil of them for a prey. A copy of the writing that the decree should be given out in every province was published unto all the peoples, that they should be ready against that day. The posts went forth in haste by the king's commandment, and the decree was given out in Shushan, the palace. And the king and Haman sat down to drink, but the city of Shushan was perplexed. Now, when Mordecai knew all that was done, Mordecai rent his clothes and put on satcloth with ashes, and went out into the midst of the city, and cried with a loud and bitter cry. And he came even before the king's gate, for none might enter within the king's gate clothed with satcloth. And in every province, whither so ever the king's commandment and his decree came, there was great mourning among the Jews, and fasting and weeping and wailing, and many lay in satcloth and ashes. And Esther's maidens and her chamberlains came and told it her, and the queen was exceedingly grieved, and she sent Raymond to clothe Mordecai and to take his satcloth from off him, but he received it not. Then called Esther for Hathak, one of the king's chamberlains, whom he had appointed to attend upon her, and charged him to go to Mordecai to know what this was and why it was. So Hathak went forth to Mordecai, unto the broad place of the city, which was before the king's gate, and Mordecai told him of all that had happened unto him, and the exact sum of the money that Haman had promised to pay the king's treasuries for the Jews to destroy them. Also he gave him the copy of the writing of the decree that was given out in Shushen to destroy them, to show it unto Esther, and to declare it unto her, and to charge her that she should go in unto the king, to make supplication unto him, and to make request before him for her people. And Hathak came and told Esther the words of Mordecai. Then Esther spake unto Hathak, and gave him a message unto Mordecai, saying, all the king's servants and the people of the king's provinces do know that whosoever whether man or woman shall come unto the king and to the inner court who is not called. There is one law for him that he be put to death, except those to whom the king shall hold out the golden scepter that he may live. But I have not been called to come in unto the king these thirty days. And they told to Mordecai Esther's words. Then Mordecai bade them return answer unto Esther. Think not with thyself, that thou shalt escape in the king's house more than all the Jews, for if thou altogether holdest thy peace at this time, then will relief and deliverance arise to the Jews from another place. But thou and thy father's house will perish. And who knoweth whether thou art not come to the kingdom for such a time as this? Then Esther bade them return answer unto Mordecai. Go gather together all the Jews that are present in Shushen, and fast ye for me, and neither eat nor drink three days, night or day. I also and my maidens will fast in like manner, and so will I go in unto the king, which is not according to the law. And if I perish, I perish. So Mordecai went his way, and did according to all that Esther had commanded him. End of chapter 4. Chapter 5 Now it came to pass on the third day that Esther put on her royal apparel, and stood in the inner court of the king's house, over against the king's house, and the king sat upon his royal throne in the royal house, over against the entrance of the house. And it was so when the king saw Esther, the queen standing in the court, that she obtained favor in his sight, and the king held out to Esther the golden scepter that was in his hand. So Esther drew near, and touched the top of the scepter. Then said the king under her, What wilt thou, Queen Esther? And what is thy request? It shall be given thee, even to the half of the kingdom. And Esther said, If it seemed good unto the king, let the king and Haman come this day unto the banquet that I have prepared for him. Then the king said, Cause Haman to make haste, that it may be done as Esther hath set. So the king and Haman came to the banquet that Esther had prepared. And the king said unto Esther at the banquet of wine, What is thy petition, and it shall be granted thee? And what is thy request, even to the half of the kingdom it shall be performed? Then answered Esther and said, My petition, and my request, is, if I have found favor in the sight of the king, and if it pleased the king, to grant my petition, and to perform my request, let the king and Haman come to the banquet that I shall prepare for them, and I will do to-morrow, as the king hath said. Then went Haman forth that day, joyful and glad of heart. But when Haman saw Mordecai in the king's gate, that he stood not up nor moved for him, he was filled with wrath against Mordecai. Nevertheless, Haman refrained himself and went home, and he sent and fetched his friends in Zeresh, his wife. And Haman recounted unto them the glory of his riches, and the multitude of his children, and all the things wherein the king had promoted him, and how he had advanced him above the princes and servants of the king. Haman said moreover, Yea, Esther the queen did let no man come in with the king unto the banquet that she had prepared, but myself, and to-morrow also am I invited by her together with the king. Yet all this availeth me nothing so long as I see Mordecai the Jew sitting at the king's gate. Then said Zeresh, his wife, and all his friends unto him. Let a gallows be made fifty cubits high, and in the morning speak thou unto the king, that Mordecai may be hanged thereon. Then go thou and merrily with the king unto the banquet. And the thing pleased Haman, and he caused the gallows to be made. End of chapter 5 Chapters 6-10 of Esther, American Standard Version. This is a Librivox recording. All Librivox recordings are in the public domain. For more information and to find out how you can volunteer, please visit Librivox.org. Recording by Sam Stinson. Chapter 6 On that night could not the king sleep. And he commanded to bring the book of records of the chronicles, and they were read before the king. And it was found written that Mordecai had told of Big Thanah and Teresh, two of the king's chamberlains, of those that kept the threshold, who had sought to lay hands on the king of Hazris. And the king said, What honor and dignity hath Ben bestowed on Mordecai for this? Then said the king's servants that ministered unto him. There's nothing done for him. And the king said, Who is in the court? Now Haman was come into the outward court of the king's house to speak unto the king to hang Mordecai on the gallows that he had prepared for him. And the king's servants said unto him, Behold, Haman standeth in the court. And the king said, Let him come in. So Haman came in. And the king said unto him, What shall be done unto the man whom the king delighteth to honor? Now Haman said in his heart, To whom would the king delight to do honor more than to myself? And Haman said unto the king, For the man whom the king delighteth to honor, Let royal apparel be brought, Which the king useth to wear, And the horse that the king righteth upon, And on the head of which a crowned royal is set. And let the apparel and the horse be delivered to the hand of one of the king's most noble princes, That they may array the man therewith whom the king delighteth to honor, And cause him to ride on horseback through the street of the city, And proclaim before him, Thus shall it be done to the man whom the king delighteth to honor. Then the king said to Haman, Make haste and take the apparel and the horse as thou hast set, And do even so to Mordecai the Jew, That citeth at the king's gate, Let nothing fail of all that thou hast spoken. And took Haman the apparel and the horse, And arrayed Mordecai, And caused him to ride through the street of the city, And proclaimed before him, Thus shall it be done unto the man whom the king delighteth to honor. And Mordecai came again to the king's gate. But Haman hasteth to his house, mourning and having his head covered. And Haman recounted unto Zeresh his wife and all his friends everything that had befallen him. Then said his wise men in Zeresh his wife unto him, If Mordecai before whom thou hast begun to fall, Be of the seed of the Jews, Thou shalt not prevail against him, But shalt surely fall before him. While they were yet talking with him, came the king's chamberlains, and hasted to bring Haman unto the banquet that Esther had prepared. So the king and Haman came to banquet with Esther the queen, and the king said again unto Esther, On the second day at the banquet of wine, What is thy petition, Queen Esther, And it shall be granted thee, And what is thy request, Even to the half of the kingdom it shall be performed? Then Esther the queen answered and said, If I have found favor in thy sight, O king, And if it please the king, Let my life be given me at my petition, And my people at my request. For we are sold, I and my people, To be destroyed, to be slain, and to perish. But if we had been sold for bond men and bond women, I had held my peace. Although the adversary could not have compensated for the king's damage. Then spake the king of Hasris and said unto Esther the queen, Who is he, and where is he, That durst presume in his heart to do so? And Esther said, An adversary and an enemy, Even this wicked Haman. Then Haman was afraid before the king and the queen, And the king arose in his wrath from the banquet of wine, And went into the palace garden. And Haman stood up to make request for his life to Esther the queen, For he saw that there was evil determined against him by the king. Then the king returned out of the palace garden into the place of the banquet of wine, And Haman was fallen upon the couch where on Esther was. Then said the king, Will he even force the queen before me in the house? As the word went out of the king's mouth, they covered Haman's face. Then said Harbona, one of the chamberlains that were before the king. Behold also the gallows fifty cubits high which Haman hath made for Mordecai, Who spake good for the king, standeth in the house of Haman. And the king said, Hang him thereon. So they hanged Haman on the gallows that he had prepared for Mordecai. Then was the king's wrath pacified. End of Chapter 7 Chapter 8 On that day did the king Ahasuerus give the house of Haman the jew's enemy unto Esther the queen. And Mordecai came before the king, for Esther had told what he was unto her. And the king took off his ring, which he had taken from Haman, And gave it unto Mordecai. And Esther set Mordecai over the house of Haman. And Esther spake yet again before the king, and fell down at his feet, And besought him with tears to put away the mischief of Haman the Agagite, And his device that he had devised against the jew's. Then the king held out to Esther the golden scepter. So Esther arose and stood before the king, and she said, If it please the king, and if I have found favor in his sight, And the thing seem right before the king, and I be pleasing in his eyes, Let it be written to reverse the letters devised by Haman, The son of Ham Med Atha, the Agagite, which he wrote to destroy the jew's, That are in all the king's provinces. For how can I endure to see the evil that shall come unto my people? Or how can I endure to see the destruction of my kindred? Then the king Ahasuerus said unto Esther the queen, and to Mordecai the jew. Behold, I have given Esther the house of Haman, And him they have hanged upon the gallows, Because he laid his hand upon the jew's. Write ye also to the jew's, as it pleaseth you, In the king's name, and seal it with the king's ring, For the writing which is written in the king's name, And sealed with the king's ring, may no man reverse. Then were the king's scribes called at that time, In the third month, Sivan, on the three and twentieth day thereof, And it was written according to all that Mordecai commanded unto the jews, And to the satraps, and the governors and princes of the provinces, Which are from India unto Ethiopia, A hundred and twenty and seven provinces, Unto every province according to the writing thereof, And unto every people after their language, And to the jews according to their writing, And according to their language. And he wrote in the name of king Ahasuerus, And sealed it with the king's ring, And sent letters by posts on horseback, Writing on swiss steeds that were used in the king's service, Bread of the stud.  Upon one day, and all the provinces of king Ahasuerus, Namely upon the thirteenth day of the twelfth month, Which is the month Adar. A copy of the writing that the decree should be given out in every province Was published unto all the peoples, And that the jews should be ready against that day To avenge themselves on their enemies. So the posts that wrote upon swiss steeds That were used in the king's service went out, Being hastened and pressed on by the king's commandment, And the decree was given out in Shushan, the palace. And Mordecai went forth from the presence of the king In royal apparel of blue and white, And with a great crown of gold, And with a robe of fine linen and purple, And the city of Shushan shouted and was glad. The jews had light and gladness and joy and honor, And in every province and in every city, Whither so ever the king's commandment and his decree came, The jews had gladness and joy, a feast and a good day, And many from among the peoples of the land became jews, For the fear of the jews was fallen upon them. Now, in the twelfth month, Which is the month Adar, on the thirteenth day of the same, When the king's commandment and his decree Drew near to be put in execution, On the day that the enemies of the jews Hoped to have rule over them, Whereas it was turned to the contrary, That the jews had rule over them that hated them. The jews gathered themselves together in their cities Throughout all the provinces of the king Ahasuerus, To lay hand on such as sought their hurt, And no man could withstand them, For the fear of them was fallen upon all the peoples, And all the princes of the provinces, And the satraps and the governors, And they that did the king's business helped the jews, Because the fear of Mordecai was fallen upon them. For Mordecai was great in the king's house, And his fame went forth throughout all the provinces, For the man Mordecai waxed greater and greater, And the jews smote all their enemies With the stroke of the sword, And with slaughter and destruction, And did what they would unto them that hated them. And in Chuchin the palace, The jews slew and destroyed five hundred men, In Parshandatha, in Delfin, in Aspatha, in Poratha, In Adalia, in Aridatha, in Parmeshta, In Arisai, in Aridai, in Vaizatha, The ten sons of Haman, The son of Hammedatha, the jews enemy, slew they, But on the spoil they laid not their hand. On that day the number of those that were slain in Chuchin the palace Was brought before the king. And the king said unto Esther the queen, The jews have slain and destroyed five hundred men in Chuchin the palace, And the ten sons of Haman, What then have they done in the rest of the king's provinces? Now what is thy petition, and it shall be granted thee? Or what is thy request further, and it shall be done? Then said Esther, If it please the king, let it be granted to the jews, That our in Chuchin to do tomorrow also according unto this day's decree, And let Haman's ten sons be hanged upon the gallows. And the king commanded it so to be done, And a decree was given out in Chuchin, And they hanged Haman's ten sons. And the jews that were in Chuchin gathered themselves together On the fourteenth day also of the month Adar, And slew three hundred men in Chuchin. But on the spoil they laid not their hand.  But on the spoil they laid not their hand. This was done on the thirteenth day of the month Adar, And on the fourteenth day of the same they rested, And made it a day of feasting and gladness. But the jews that were in Chuchin assembled together On the thirteenth day thereof, And on the fourteenth thereof, And on the fifteenth day of the same they rested, And made it a day of feasting and gladness. Therefore do the jews of the villages that dwell in the unwalled towns Make the fourteenth day of the month Adar A day of gladness and feasting, And a good day, and of sending portions one to another. And Mordecai wrote these things, And sent letters unto all the jews That were in all the provinces of the king Ahasuerus, Both nigh and far, to enjoin them That they should keep the fourteenth day of the month Adar, And the fifteenth day of the same, yearly. As the days wherein the jews had rest from their enemies, And the month which was turned unto them From sorrow to gladness, And from mourning into a good day, That they should make them days of feasting and gladness, And of sending portions one to another, And gifts to the poor.  But when the matter came before the king, He commanded by letters that his wicked device, Which he had devised against the jews, Should return upon his own head, And that he and his sons should be hanged on the gallows. Wherefore they called these days, Purim, after the name Purr. Therefore, because of all the words of this letter, And of that which they had seen concerning this matter, And that which had come unto them, The jews ordained and took upon them, And upon their seed, And upon all such as joined themselves unto them, So that it should not fail, That they would keep these two days according to the writing thereof, And according to the appointed time thereof, Every year, And that these days should be remembered and kept Throughout every generation, every family, Every province, and every city. And that these days of Purim Should not fail from among the jews, Nor the remembrance of them perish from their seed. Then Esther the queen, the daughter of Abihail And Mordecai the jew, Wrote with all authority to confirm this second letter of Purim, And he sent letters unto all the jews, To the hundred twenty and seven provinces of the kingdom of Ahasuerus, With words of peace and truth, To confirm these days of Purim in their appointed times, According as Mordecai the jew and Esther the queen had enjoined them, And as they had ordained for themselves, And for their seed, In the matter of the fastings and their cry. And the commandment of Esther Confirmed these matters of Purim, And it was written in the book. End of chapter 9 Chapter 10 And the king Ahasuerus laid a tribute upon the land, And upon the aisles of the sea, And all the acts of his power and of his might, And the full account of the greatness of Mordecai, Where unto the king advanced him, Are they not written in the book of the chronicles Of the kings of media and Persia? For Mordecai the jew was next unto king Ahasuerus, And great among the jews, And accepted of the multitude of his brethren, Seeking the good of his people, And speaking peace to all his seed. End of chapter 10 And also the end of Esther, American Standard Version


Super Bowl 56 taking place this Sunday right here at SoFi Stadium in Inglewood. This year Super Bowl once again designated as a tier one level security event with more than 40 federal, state and local agencies including Cal OES participating. But the message for the general public, if you are coming to any of the events surrounding the Super Bowl this week, be on alert. If you see anything that looks suspicious or out of place, say something. It is not only that federal, state, local and private sector partner together to ensure a safe and secure Super Bowl, but we also need the public's vigilance. And that's why we say if you see something, say something. I pay attention for 11. I watch plays and make important calls. While I'm busy watching the game, I need your help keeping an eye on the stands. We all play a role in protecting our communities. If you see something suspicious, make the right call. Tell the law enforcement official. We've got more information on public safety and the see something say something campaign. You can find it on our website at CalOES.ca.gov.


I'm Brian May in the Cal OES newsroom with a quick update on these unprecedented times in California and abroad. A series of hurricanes followed almost immediately by a devastating 7.1 earthquake in Mexico. And through it all, California has been there to help our neighbors. California has now sent search and rescue teams to help with hurricanes Harvey, Irma and Maria, as well as with the earthquake in Mexico. These teams being deployed from locations up and down the state. These are really specialized resources that exist in California. It is a tremendous asset. And so we're happy to send out these resources when the need occurs, but let's be clear when we have a big earthquake here in California or a major disaster, we're going to be turning to our other states, to our federal government, and asking for assistance. So this is really a mutual aid partner, helping partner, one team, one fight effort to be able to address what the needs are. So if you'd like to help, and there are many people who have asked, there are several ways that you can help those in need. And the quickest and easiest way is just by donating money to a trusted organization. California Volunteers has pulled together some of the best and easiest options for you to do that. You can find links on their website. You can also find their link and more information on our website, oesnews.com.


So, last lecture we started our study of linear maps, in particular two-dimensional maps, and for the moment we focused on the case in which the eigenvalues were real, which in itself is fairly special case, even though it's important. Now today we will look at the case in which the eigenvalues are complex, so we start with the proposition straight away, if A2 to R2 is an invertible linear map, eigenvalues alpha plus or minus i beta, with beta difference from zero, so as you know when you have complex eigenvalues then you have, they always come in pairs, matching as conjugate eigenvalues, it is linearly conjugate to the matrix alpha beta minus beta alpha. So this matrix here is in some sense the diagonal form, or the canonical form of matrices with complex eigenvalues, right? So I think, actually I will skip the proof of this proposition, it's a basic result of linear algebra, and I rather want to concentrate on the analysis of these. So linearly conjugate of course means in the same way as we were talking about before, so there exists P linear map such that, so if we call, let's call this B, okay, such that A composed with P is equal to P composed with B. So in particular this allows us to, this defines the linear conjugacy classes as we, in the lemma that we talked about the other day, if two maps, if two linear maps are linearly conjugate then they have the same eigenvalues, okay? So this means that if you take two maps which have different eigenvalues then they will be linearly conjugate to different linear maps, and so these two linear maps cannot be, these two maps cannot be linearly conjugate amongst themselves, sorry? Yes. So using this linear conjugacy we can study the, we can reduce the study of all linear maps up to linear conjugacy to the study of this family of maps. So let's try to understand what is the dynamics of this family of maps. So what is the dynamics of linear maps of this form? Well we have a map from R2 to R2. So the easiest way to understand is actually to identify, we can identify the space R2 with the complex plane in the sense that if we take any points x, y, we identify this with the point x plus i, y, right? So we think of this map, we think of this space R2 as the complex plane where each coordinate, so this is the imaginary axis and this is the real axis, and why do we do this? Because then there is a very simple form for this expression and it's the following. So in this case then the action, so let me call this okay b just to be consistent with what we have, then the action of b, action of b corresponds exactly to duplication by alpha minus i beta. So what do I mean by this? I mean that if I take, if I apply this matrix x, y, sorry this matrix v, sorry if I take a point here, okay this is a vector v, let me call this v1, let me use the same notation that I've been using before, v1, v2 are the two coordinates, right? So b of v1, v2 is equal to alpha beta minus beta alpha v1 v2, okay and this of course is just equal to alpha v1 plus beta v2 minus beta v1 plus alpha v2, right? This is the image of the vector v1 v2 under the matrix b. Now if we think of this vector, if we write this vector as a complex number, if we write v, so this is b of v, so if we write v as equal to v1 plus i v2, so it's just a slightly different way of writing this vector, right? You write this with the notation of complex numbers and then this is exactly equal to, then we have that v1 plus i v2 times alpha minus i beta, okay? We now multiply them as complex numbers or maybe, sorry let me write it to be more consistent with what we have above, of course it's the same thing, all right? Alpha minus i beta times v1 v2, so what does this give? This gives alpha v1 minus i v1 beta plus i v2 alpha plus beta v2, which is equal to alpha v1 plus beta v2 plus, so v2 alpha minus v1 beta, which you can see is exactly the same as what we have above, right? So the real part which is the horizontal coordinate is alpha v1 plus beta v2 and the imaginary part is minus v1 beta plus v2 alpha, it's exactly that. So this is remarkable in some sense, it's like a little magic. So what we're saying is that this, what in disguise, this is a completely real linear map from R2 to R2, okay, and it acts on vectors in a certain way that is not so easy to visualize, okay? Because you take a vector v, you take v1 to v2, the new horizontal coordinate of the image of v is given by this formula which is some kind of composition of v1 and v2. The vertical coordinate is given by this formula, it's not immediately easy to see what this map does. But by identifying this with a complex plane, it turns out that this action can be written exactly as multiplication by this complex number. So you take the, if it has a complex number, you multiply by a complex number and you get the image of this and the linear map. Why is this useful? Because multiplication of complex numbers is something we can, we understand fairly well, right? So what is the best way to multiply complex numbers? So what the iteration, okay, so bn of v is equal to alpha minus i beta n times v, right? Times v1 plus i v2. This is what the iterates are. So all you're doing when you iterate this map is you're multiplying again and again by this complex number, right? So does this help us to understand what this, remember what we're trying to understand is what this vector does, right? What is the omega limit of, we're trying to understand the dynamic. So we're trying to take a vector, iterate it under the map and see what it does. Does it go to infinity? Does it go to zero? What does it do? So how do we find out what this does? So it boils down to what the iterates of, this is one of the eigenvalues, right? So really the future forward orbit of this vector here is basically determined by these iterates of these eigenvalues, the products of these eigenvalues. So what are the products of these eigenvalues? Anyone have any idea how to understand what's happening here? Exactly. What's the best way to multiply complex numbers? It's much better to multiply in polar code, that's exactly, okay? So, and once you have in polar code it's very easy to multiply and we see exactly what happens. So write, we'll use the notation, write alpha minus i beta is equal to R E i theta in polar coordinates where R is equal to the modulus, right? And theta is equal to cosine minus one of alpha over alpha. So this is the formula that translates the standard coordinates in the complex notation to the polar coordinates. So in this notation we have that BN of V is equal to R, so how do you multiply polar coordinates? You know that what you do is you just multiply the, whatever this is called here, the modulus, right? You get R to the N and then you get E to the i N theta, right? Times this vector in polar coordinates, right? So this we can write as some R0 E to the i theta 0 where this here is the initial condition in polar coordinates also. So we write this point as some R0 E to the i theta 0, but of course I0, this is R0, this is theta 0, so this defines this point here. So now we can basically read out exactly what the, what the image is, right? So BN of V is equal to R0 EI theta 0, B of V is equal to RN EI theta N times R0 EI theta 0, okay? So what happens to this? Sorry? Wait. Mario. So this is equal to R0 times RN times E to the i theta 0 plus theta N. This is the iterates, right? And what do these look like? What is this point doing? Okay, good. So let's look at the modulus first. What is this modulus doing? Is this point converging to 0? Is it converging to infinity? Is it doing something strange? It depends on whether R is less than 1 or bigger than 1, okay? Notice also in this case the point 1 is a crucial dividing point, right? And what is R? R is exactly the modulus of the eigenvalue, right? So it depends on the eigenvalue. So if these are the eigenvalues, so this is the unit circle 1, right? So if alpha minus i beta is inside the unit circle, then that means that the modulus R would be less than 1, yeah? And so what will happen to this point? The distance from the origin will shrink to 0 as you iterate in forward time. And what happens to this, how does it shrink to 0? How does the actual point move? As you remarked, what happens is that at every iteration you rotate by an angle theta. So it shrinks in a rotation. So the orbit of V is contained in the parameterized curve R0, R to the t times EI theta 0 plus theta t, t in R, okay? Just like in the real case, I said you can think if there is a parameterized curve in your space and so what does this curve look like? So this is the eigenvalue space, eigenvalues, okay? And this is the dynamical space. So here you start with some point R0, E to the i theta 0, okay? And what happens is that you are on this parameterized curve so you see that when t is equal to 0, the same as when n is equal to 0, t and n is the same. When t is equal to 0, you get exactly R0, EI theta 0. Then as t increases, if you take t continuous like this, then it will slowly, it will slowly rotate by an angle theta t. So when t is small, it will be a small angle and it will slowly shrink, okay, by this factor. So what you get is slowly shrinking curve. The actual orbit will be discrete set of points, but it will belong to this curve because when we're taking the orbits, we're just taking the integer values. So this will be R0, this will be what we call V0, this will be V1, this will be V2. At each step, the rotation will be constant because the rotation is always by an angle theta, right?  So of course, how fast it shrinks depends on the value of R. If R is very close to 0, then it shrinks very fast, right? So it might be, the picture might look for example like this. It's still spiraling, but it's just spiraling very fast. If R is very close to 1, then it is shrinking very slowly. So this makes sense because this is again, if the eigenvalues have inside the unit circle. So remember, the two eigenvalues are complex conjugates, so if one of them is inside the unit circle, then the other one is also inside the unit circle. What happens if the eigenvalues are outside the unit circle? What happens if the eigenvalues are outside the unit circle? Exactly. If the eigenvalues are outside, then we have that R is bigger than 1, okay? If eigenvalues, if R is bigger than 1, then you just have exactly the same picture, but the spiral in forward time goes out, okay? Of course, in backward time it goes in, right? There's a symmetry, exact symmetry. So this spiral exists for all T and R, so for positive values of time and negative values of time. For negative values of time, this is spiraling outwards. So it's a completely symmetric situation here in forward time and in backward time. What happens if the eigenvalues are on the unit circle? The distance R is equal to 1 in that case, but you still get a rotation, right? So if R is equal to 1, then what is the picture that you have? You have one, you have your point v0 here, and if you look at this curve, this is just constant distance from R0, e to the i theta 0 plus theta t, so it's a constant, so it's a circle, okay? It's not hyperbolic, exactly. And we will see later what the significance is. This is very significant, there's a very big difference. Well, you can immediately see a big difference. So this is clearly the complexified version of having, no, it's a version of the non-hyperbolic case with real eigenvalues, right? In those cases, you had that the, every point is a fixed point, okay? You no longer have that every point is a fixed point, but you have that every point lies on a circle. So if you start on this circle at a certain distance from the origin, you stay at that same distance all the time. So you have all these invariant circles, and we will study later on next week the dynamics that happens with this important invariant circle, which is also very rich and very interesting. But this is the non-hyperbolic case. So for the moment, we're not going to worry too much about this case. Okay, so these are really the way you understand the dynamics. It's very simple. Before we go on to talk about topological conjugacy, let me remind you that these are the pictures that you get for the normal form, right? For matrices of this form. But we said that in general, if you have a matrix with complex eigenvalues, it is linearly conjugate to this form. So what would the picture look like in the general case? There is not that form. If you take another map that is linearly conjugate. So in general, if A equals ABCD, okay, has eigenvalues alpha plus or minus i beta with beta different from zero, because otherwise it would just be real eigenvalue, then what is the picture? Well, we know that it's linearly conjugate to alpha beta minus beta alpha. But what does the picture look like? The certain things which we know have to be preserved, right? In particular, clearly a linear conjugate is a special form of topological conjugacy. So you must preserve the fact that if these are inside the unit circle, then you get this picture where everything converges to zero, and therefore here you must somehow also converge to zero. Right? So if you take some vector v zero, it must somehow converge to zero. The linear conjugacy says quite a bit more, it must converge to zero. So there's a linear change of coordinates if you want that maps this to this, right? So in fact, what the picture is in here will be very similar to this, but basically it's the same thing as happens in the case with real eigenvalues. What it can, the only thing that can happen is that it can be squashed in certain directions, right? Because that's what the linear conjugacy does. The linear conjugacy basically, because it's a linear change of coordinates, there's not that much strange things it can do. It basically can just squash it, it's a linear map, so all it can do is map a circle to an ellipse and kind of squash it in a certain way. So for example, it may look like this. So it's still converging to the origin, a key difference that in fact is one of the reasons why it's much more useful to start in a normal form is that it does not converge monotonically to the origin, right? So we will use now in a second the fact that in this case, we converge monotonically to the origin in the sense that if you start at a certain point and you're converging to the origin at every other point, you're closer and closer monotonically to the origin. Whereas notice that this is not the case here, right? Because you start at a certain distance from the origin, then you come closer, suppose this is the image, here you're at a certain distance, but then you map, for example, to here, which is further away from where you were.  Every time, you're slowly moving forward, but it's not monoton, right? Closer, a little bit further, a little bit closer, a little bit further, but on the whole, on average, getting closer, okay? And this makes a difference to how we study. The good thing is that every time we have this kind of picture, we can find a linear conjugacy that maps it to that case, and that's what we're going to do. So the next step is a topological conjugacy, okay? So suppose we have two maps with different eigenvalues. So suppose we have a prime to such maps topologically conjugate. For example, if we have a here, a b c d, a prime here, a prime, b prime, c prime, b prime, and suppose that they both have complex conjugate eigenvalues, and suppose that the corresponding pictures, so one of them maybe converges quite slowly to zero like this, okay? And let's suppose that the other one, so if one has eigenvalues inside the unit circle, so suppose lambda is less than one, and suppose this one has lambda greater than one, can they be topologically conjugate? So wait, so here alpha plus i beta plus or minus i beta less than one, here alpha plus or minus, alpha prime plus or minus i beta prime greater than one, can these be topologically conjugate? No, why not? Different omega limits, right? In one case everything's converging to zero, in forward time, in the other case everyone's converging to infinity. So let's suppose both of them have less than one, okay? But they look different. So for example, this could be like this, and the other one might have a different shape and it might go very, very fast to zero. Might these be topologically conjugate? Different eigenvalues, so they cannot be linearly conjugate, but as we've seen before topological conjugacy is a much weaker condition. Is there any immediate obstruction to the fact that they're topologically, sorry? The rotation, they might have a different rotation, they might have different, they might have different, in principle, in principle. What about if this actually has real eigenvalues? So let's suppose that this has real eigenvalues, so let's suppose this has, the picture here is like one of the cases we studied before, and actually the picture looks like this. Do you think that something with real eigenvalues could be topologically conjugate to something with complex eigenvalues? Look very different. But they both have an attracting fixed point, globally attracting fixed point, so they both have a fixed point that attracts anything. Okay, so what's the theorem going to be? So theorem two hyperbolic invertible linear maps with distinct real or complex eigenvalues. So if they're complex, they're automatically distinct, right? I'm just emphasizing that we are not restricting ourselves to complex eigenvalues, real eigenvalues in general with distinct eigenvalues, topologically conjugate, if and only if their fixed points are both of the same type, both of the same type, by type I mean either attracting or repelling or subtle, they, the same index by which I mean the number, number eigenvalues with positive real part. So the most important part of this theorem is the first part that says in particular that these two systems are topologically conjugate. So forget about the distinction. On a topological level, complex eigenvalues, real eigenvalues doesn't make any difference. If they're attracting, if they're hyperbolic, right, we're assuming they're hyperbolic linear maps, then if the fixed point is attracting, they're topologically conjugate, including this case and this case, right? So what this means is that this, what we're going to have to show to do, remember that a topological conjugacy has to map orbits to orbits. So basically it means can we find a homomorphism that maps this spiral to one of these curves is essentially what we're going to do. So that is why intuitively it seems so strange that this would happen because you could kind of tell that for this to be a homomorphism, you need to be able to do that. And it seems intuitively not obvious that you can map this spiral that goes down and around to this curve. So that's the most important part of this. The second part is just a requirement basically related to what you were saying about the orientation because you can have some cases in which they're both attracting, but one of them has, I can't think now of the different cases, but you can have the number of eigenvalues positive real part also relates to the number of eigenvalues with negative real parts. And you remember if you have one with positive real part and one with negative real part, then it's orientation reversing, otherwise it's orientation preserving. So this is basically a requirement that takes care of the fact that if they have to have the same orientation, then this is an additional requirement. This is in some sense a kind of technical requirement. The point that we want to emphasize more is this. But of course the saddle type can only occur if the eigenvalues are real. So we have already proved that part, right? So if they have the same, if they are both saddle points and they have this property, they have the same index, then it's, we've already discussed the topological conjugacy when you have real eigenvalues. So really the part we need to concentrate on is this in the case in which they're attracting or repelling, the two cases are completely symmetric, as we said, because one is just attracting forward time, one is for attracting backward time. So they are symmetric. So really we will just prove this theorem for the case of two invertible hyperbolic linear maps with attracting fixed points and everything else is taking care of. And I think just for simplicity, let us choose this particular configuration. We could do it with two maps, both of which have complex eigenvalues. But it's exactly the same argument, okay? So for generality and to emphasize the fact that these are included also, we will carry out the construction in this case. Okay, so let's do that now, yeah? So we need to, like we've done before, we need to construct this conjugacy. So what's the first thing that we can do? So here, in fact, I didn't even draw the general case. I remember that this is already the diagonal case. So the general case, you would actually have something a little bit even more complicated if you remember you had the two eigenspaces. And the picture was more something like this. So they would anyway be always come in tangent to a certain direction, depending on the specific eigenvalues that you've got. So this is actually the more general picture in the case in which you have real eigenvalues and an attack, hyperbolic attack at fixed point. And this is the general case with hyperbolic. So the first comment, just like in the real case, is that by linear conjugacies, each of these is linearly conjugate to its own canonical form, right? This is linearly conjugate to a matrix of the form alpha beta minus beta alpha. And this is linearly conjugate to a diagonal form. So the first thing we do is we put them in their normal forms so that we study this. So proof, a and a prime, so a, a and a prime, so suppose, okay. So let's suppose a equals a, b, c, d has eigenvalues alpha plus or minus i beta. With beta different from 0 and alpha plus or minus i beta less than 1. Just for deafness. And i prime equals a prime, b prime, c prime, d prime. Eigenvalues lambda 1 and lambda 2, lambda 1, lambda 1 different from lambda 2. Distinct eigenvalues, lambda 1 and lambda 2 in R. And lambda 1 less than 1 and lambda 2 less than 1, okay? This is just a specific case. We're going to assume that it's attracting. The other cases are all done in exactly the same way. So then a is linearly conjugate to alpha beta minus beta alpha. And a prime is linearly conjugate to lambda 1, 0, 0. So it's enough to show that these two maps are topologically conjugate, right? Because that by linear conjugacy is in particular topological conjugacy. So by the properties of the equivalence relation that would imply that a and i prime. So let's just, for simplicity, let's just call this one also a. Let's just assume this is a and let's assume this is a prime. And we just show that these are topologically conjugate. So how are we going to do it? We're going to use the technique that we know, which is the technique of fundamental domains. So what is the picture? And this is where we use the fact that these are normal forms. So the picture here is that we have this spiraling inwards like this. And the picture here is that. So are we going to be able to find a homomorphism h that maps these two pictures to each other? So the trick here is not to get scared. We can do this. So we're going to find a fundamental domain for both maps. And then we're going to map the fundamental domains by homomorphism and use exactly the same technique that we used in the one dimensional case. So how do we find a fundamental domain? Let's do it for this one first. Maybe it's a little bit easier to visualize this fundamental domain. What do you think is going to be the fundamental domain here? An annulus, good one. Where shall we put the annulus? Yeah, yeah, yeah, whatever we want. So for example, let's take the unit circle, but any circle will do. That's one boundary of the annulus. How do we define the other boundary of the annulus? We map the circle, very good. So we take the unit circle and then we take the image of the circle. Now, how do we know that the image of the circle is strictly contained inside itself? So in this picture we know because every point maps closer to the origin. So in this case, the image of the circle will be something like, let's see, it will not be a circle because in this case we are mapping the horizontal is stronger. So I think it would be something like this and ellipse like this. The image by the linear map because that is what. So each of these points, so this point maps to this point, this point maps to this point, this point maps to this point, this point maps to this point. These curves, these invariant curves, they define exactly where each point on the outer circle maps to the inner circle. And our claim is that this region here should be a fundamental domain. So let's write this a little bit more formally. So it's like S equals unit circle. Consider D equals the annulus bounded by S. And so this map is what we called A and this map is what we call A prime. So A prime of S. So this here is S. This is A prime of S. So lemma. So let's do the same thing on the other side. We can do the same thing. Let's do all at the same time the two fundamental domains. So here we can also take the unit circle here and what will be the image of the unit circle. In this case it will actually be a circle in this case because you're just rotating and shrinking uniformly in every direction. So you just get a smaller circle. So here this is also the unit circle and this here is A prime, sorry, A of S. And this is the region in this case, this is the annulus. And again, notice that we have these curves are telling you where these points are mapping, right? So this inside interior boundary of the annulus is defined by taking the image of S and in particular this point maps to here, right? And then you have here I just do one spiral. Here I do many of such curves. But also here you have spirals going through, you have many spirals. So for this initial condition, you have a spiral doing this. Okay, it's a bit more difficult to draw. That's why I didn't draw it, okay? But this point maps to this point. So every point on the outside of this annulus rotates and shrinks and maps to some point on the inside. So there's a bijection given by these spirals between the outside annulus and inside. Okay, so similarly we define D prime. So D prime is the annulus bounded by S and A prime of S. So lemma is that D and D prime are fundamental domains for A and A prime. On what? Not exactly on the whole space R2. Sorry, excuse me. The annulus, the annulus bounded by S and A of S. Annulus bounded by the boundaries are S and A of S, sorry here. S and A of S and here is S and A prime of S. The blue region, it's the unit. So we can either call them S prime or S, it doesn't really matter. If you want we can call it S prime. Is that what's following you? This here I'm mapping A prime, so S and A prime, okay? So these are fundamental domains but they cannot be for everything, right? So remember the origin of course is a fixed point. So the origin is the only point that is fixed in both forward and backward times so it has no chance of entering this fundamental domain. This is just a small remark. But of course these are fundamental domains at most for everything except the origin. So how do we prove this lemma? Well, it's not difficult, right? Because it is really just again the one-dimensional argument because what you want is because every point lies on one of these curves then really it is sufficient to show that this is a fundamental domain on each curve, defines a fundamental domain on each curve. So for any initial condition that you choose here it's orbit in forward and backward time will lie on this invariant curve, right? So this invariant curve is really just a one-dimensional curve and so the argument is exactly the same as we used in the one-dimensional case. So if you are on the outside of this annulus then in forward time you will at some point you must meet the annulus. Why must you meet the annulus? Because this is a point here, this is the image of this point. So because the dynamics is order preserving, we assume, okay, this is a case in which it's orientation preserving. So we assume in this particular case, in the orientation reversing case you switch and you just have to kind of keep that in mind a little bit. But in the case in which the index, all the eigenvalues are positive then this is just moving along this curve and then before the dynamics is order preserving along this curve and then once you're here you cannot jump all the way to the other side of the annulus. Can you see that? You convinced? So let's try to write this down. Proof. Each 2 minus 0 lies on a curve gamma of x which is invariant under A such that A of gamma, let me write this as gamma of x, A of gamma of x equals gamma of x. You agree with this? There's a curve that is invariant, right? Each of these curves that I draw are invariant because what we said is that every point that lies on one of these curves its image lies on the curve also. So that means that if you take the whole curve and you look at its image under the map it's the curve itself. And the same for each one of these spirals. If you look at one of these spirals the orbit of every point if you start on the spiral you stay on the spiral this means that A of gamma is equal to gamma. So these curves are invariant on the dynamics. And dynamics on gamma of x is monotonically decreasing to 0 in forward time. How do we know this? Exactly. Because we've used the normal forms. This is the crucial point in which we take advantage. This is the remark I said before. We take advantage of the fact that we've reduced by linear conjugacy to the normal forms. The crucial difference between the normal forms is exactly that you are converging monotonically to 0. And in the general form if you did not have the normal form I'll make a little picture here of what the real picture looks like. The real picture looks like this. And in this case if you take the unit circle here then you might intersect and if you take the image of the unit circle it might actually not even give an annulus. So the image might not give an annulus. And also this curve intersects the unit circle in many points. Look this curve which is one of the invariant curves intersects the unit circle here, here, here, here, here in many points. After some time it eventually stays inside the unit circle all the time. But it might come in and out, in and out. Whereas when you take the linear conjugacy to the normal form then you get, as I said before, because the normal form is of the form r0r to the nEI theta 0 plus theta to the n rn is decreasing monotonically in t. So this lies on the curve t. This lies on this curve. And this distance from the origin is going to zero monotonically. So this cannot happen. You can only cross the unit circle once. And you must cross the unit circle once. The curve, this is the curve, okay? Decreasing in full time. So gamma x intersects unit circle as in a unique point. It must intersect in a unique point. Every curve, okay? So no matter what initial condition you have, whether it's outside the unit circle or inside the unit circle, it lies on an invariant curve, parametrized curve, which intersects the unit circle in a unique point. This is how we've used the normal form. It's the same here, okay? Here, in this case also, in the general setting, we have the eigen, in the general setting, we have that the eigenspaces might be like this. And so this might be like this. And these, and again, if you take the unit circle, this might, okay, I didn't do it very well, but you might have a situation where this might come like this, okay? So it might intersect the unit circle, then come out the other side and then converge again to this, okay? So in the general case, you might not, you might intersect the unit circle several times, three times in this case instead of one time. But when you straighten it out by linear conjugacy, then you must intersect only once. So moreover, the orbit is monotonically decreasing. So maybe what I wanted to emphasize here is not so much the dynamics, but gamma x is monotonic. So geometrically, gamma x is monotonically decreasing to zero in forward time, okay? It's a parameterized curve in T. And moreover, the orbit of x on gamma x, the orbit of x on gamma x is also monotone, is also monotonically decreasing in forward time. And monotonically increasing in backward time. It is sufficient to show, it is sufficient to show that for all x in R2 minus zero, there exists unique tau of x such that A tau x of x belongs to gamma x intersection D. I haven't really said anything here. I'm just trying to formalize the fact that all we need to do is look along one of these curves. So as I said, this is exactly the one-dimensional argument and how the one-dimensional argument goes. So let tau x, so suppose first, suppose we take x outside the unit circle, okay? And let tau x greater than or equal to one be the smallest integer such that A tau x minus one of x is greater than one, greater than or equal to A tau x. So you agree that such an integer exists, right? So you just look at the point x and you wait, and there will be some point when it reaches the unit circle, right? So there will be some point where at time tau x minus one, it's outside and at time tau x, it's either on the unit circle or inside the unit circle. This must happen because the point is converging to zero in forward time, okay? Then again, by the monotonicity of orbits, we have that A tau x of x is greater than, sorry, these are absolute values here, then A tau x is greater than A of x A of A of x intersect the unit circle, and this is greater than or equal to A of tau x plus one. So what I've done is I've just applied A to each of these terms, right? So this point x, the orbit of the point x lies on this curve gamma x. So I'm thinking, okay, take the first time such that A tau minus one is here, A tau is here, then this is the point of intersection gamma x of the curve with S one. So you have three points here that have a certain order on this curve, right? One is bigger, the distance from the origin has a specific order between these three points. And so now when I apply A to all three of these points, this order is preserved, right? And so what happens when this order is preserved? Well, this point here maps to A tau of x because this is A tau minus one of x and this is A tau of x. So this maps to here, this point maps to here by definition of this annulus, and therefore this point must map outside the annulus, which is what we're saying here, A tau x plus one x is less than equal to this, okay? And this is exactly the definition of the boundary of the inner boundary of the annulus. So this shows the uniqueness. This shows what is, it should be intuitive. Obviously, which is that each point must fall. So what this shows is that you must fall inside the annulus and you must fall a unique time because when you're inside the annulus, you must fall outside the annulus. There's no other alternative, unique point such that. Any questions? This is just exactly the one dimensional case. So if you have done the exercise that on the fundamental domains for the one dimensional case is exactly the same. Like this is how we prove that the interval that we choose in one dimensional case is a fundamental domain exactly in the same way. You have on the line, you have points moving back and forth and they can only map once in the fundamental domain because of exactly the same argument. So here, this curve plays the role of the real line in the one dimensional case. The restricting to this curve, we have exactly just like a one, all the properties of a one dimensional map, one dimensional linear, no it's not linear in this case but it's got exactly all the same properties and before this is exactly the same argument. So if you're finding this difficult then my suggestion is do the one dimensional case and you will see it exactly. You will understand exactly what. And you can see that this, so there's a couple of comments. First of all, this holds in both cases because we've not used anywhere the particular property of this. When you just look at the annulus if you forget about what happens inside here, outside here, if you forget about what the picture actually looks like then it's exactly the same. All the arguments are exactly the same. You just use the fact that every point lies on an invariant curve and that each of these invariant curves maps, each of these invariant curves intersects the unit circle in a unique point and that this is its image. So here, the only difference when you look at here is that these look a little bit more straight whereas these have these rotation components and this maps to this and this maps to this and this maps to this. But the argument about monotonically decreasing is all exactly the same. So there's no difference in this construction of this proof that the fundamental domain is exactly the same. The second comment is that we have to be a little bit careful about the boundaries, right? Should the boundaries be included in the fundamental domain or not? One of the boundaries should be included, okay? I did not mention that because I wanted to wait. So here, you see if you include both the boundaries, then you get that this point lies in the fundamental domain and its image lies also in the fundamental domain. This is the only point that's a little bit problematic. But if you just include one of the boundaries, then of course its image then lies outside the fundamental domain and it's no problem, okay? So this annual was just like again in the one dimensional case, remember? We took the interval half open and half closed for exactly the same reason, okay? So this is a comment and when I informally define the annual i, d and d prime, I did not say whether they included the boundaries or not. So you need to remember that you need to include one of them. You can choose whether you want to include the outer boundary or the inner boundary. It's the same. Okay, so how did we continue the one dimensional case? So now we have the fundamental domains. Remember what we need to construct this conjugacy is to find a bijection between the fundamental domains or the homomorphism between the fundamental domains. There is a little bit more complex here though than in the one dimensional case. Why is that? Because there are many, so we now construct a homomorphism. So what would be the natural homomorphism to define here?  But because we then want to extend it to a homomorphism of the whole space, remember, right? So we start with a homomorphism h tilde between the fundamental domains. Then we cannot just use anything really because so we want, so suppose we send s, we send the unit circle to the unit circle in any way we want. So for example, we could send it, use the identity, right? So for example, let h, the stricter to s equals the identity, h tilde. What is the natural way to extend this h tilde to the other points of the annulus? Okay, we want to preserve the dynamics. So what is the obvious way? So s, we can map to s. Okay, what shall we map this point to? Yes, well in this case we don't need to be so precise but we need to at least use this curve here, right? Okay, so wait a second, let me just look at my notes and see how careful we need to be here. Yes, yes, we need to, yes, so we need to map these curves to each other. So if you take any point here, right, this each annular in both cases is filled by these solution curves. So in this case, so in this case they look a bit more like this, these are the solution curves. Sorry, not the curves that are invariant by the dynamics by the orbit. So the obvious way to define h tilde is to define s, to define h tilde on s from s with can be an arbitrary homomorphism for example, the identity and then if you look at the image of some point inside here, then you look at the curve and then you look at the weather curve intersects s and you look at the corresponding point s here and then you map this curve to this curve here and that should be sufficient to do that, right? So you can define, so let me write it like this, so lemma, so let d bar and d bar d prime bar be the closures of d and d prime. So this now include both of the boundaries, okay? Then there exists homomorphism h tilde from d bar to d bar prime, such that for all x in the unit circle h tilde of x equals, so here I'm using slightly different notation, so h tilde of x is equal to a prime, a prime minus one composed with h tilde, composed with a. You take a point in s one and first you apply a, so you take a point on the outer boundary, which is s, okay? And then you apply a of x, which gives you a point on the inner boundary and then you apply h tilde and this gives you the same thing as taking the image of s and applying a prime here, right? So it's exactly the same. So we're saying that these homomorphism must match up on, they must conjugate the dynamics on the boundary. So you must map, you take s one and you take the image of s one and then you take under the homomorphism and this conjugates the dynamics on the two boundaries. All right, this is what this is. Yes, h tilde, yes, this statement here. So I'm saying that there exists a homomorphism h tilde from deba to deba, which conjugates the dynamics on the boundaries. Let me draw a better picture because these pictures have become a little better. So the properties I need is that the boundaries should match up. So this is one annulus. So this is s and this is a of s. And here I have the other annulus. So this is the annulus d, so this is the annulus d and here I have, this is a prime of s, this is d prime. So I know that to each point of s corresponds, the point, if I take a point x here, this is a of x. Is that clear how these two, yeah, these two, the inner boundary is the image of the outer boundary and it's the image by the map, by the linear map and the linear map has these invariant curves. So it's natural to think of it as coming from these invariant curves like this, right? So the outer boundary maps to the inner boundary. And here is the same thing. Here you also have these images, although they have a different shape. Doesn't matter to have these images, okay? So what I want in order, I'm trying to define h tilde and I'm trying to define it on the closure of these two, of these two annuli. And because of the way I want to extend it as a conjugacy to the whole space, I want it to be, because I can define in lots of ways a homomorphism on these two annuli, but I want to make sure that it conjugates the dynamics on the boundaries, which is the only place I can conjugate to, right? Because every point that's in the interior of the boundary after one iterated maps outside the boundary. So it wouldn't make sense at this stage to talk about conjugacy for any other points. But the points that are on the outside boundary after one iterated the map to the inside boundary. So there are points in the bar that after one iterated the way map also inside the bar, which are exactly these. And so I want this conjugacy condition to hold here. I want to make sure that I have a homomorphism from D bar to D bar, D bar to D bar prime that satisfies this property. So if I take a point X here, I look at A of X, right? And then I take H bar. So I take, suppose this for example is H bar of X, right? And then this point here is A prime of H bar of X. And I want to make sure that this is the same as H bar of A. So that this point here is the image of this point here. That is just a standard conjugacy condition. Is that what I wrote? A prime. So if I invert A, what I get is maybe I didn't like, maybe that's not what I wrote. H tilde, so what I want is this. A prime composed with H is equal to, composed with H tilde is equal to H tilde composed with A. So this, yes, yes, so this is the same as saying H tilde equals A prime minus one composed with H tilde, composed with A. That's what I had here. So I won't be able to finish, unfortunately, the construction today, but once we have this H tilde, we proceed exactly as in the other cases. We extend it in the obvious way, okay? Then let H from R2 to R2 be given by H of 0 equals 0 and H of X equals 2. So we take any point X and we use the same argument. We go all the way until we reach the fundamental domain and then we pass it by H tilde and then we iterate backwards, right? So we go A, A prime minus tau of X composed with H tilde, composed with H prime, A tau proposition. H is a homomorphism. So I will not prove that today, but that's all we need to prove because if you remember, this system automatically gives a conjugacy in an abstract setting, we've already shown that. So as long as you have two fundamental domains and you have a bijection between the fundamental domains, you can extend that bijection to the whole space by exactly this formula and this gives a conjugacy. What we need to worry about is the regularity of the conjugacy, the continuity of the conjugacy, and that is precisely why we had to choose H tilde in this specific way. Okay, so I forgot to prove this lemma. This is a simple lemma because all you need to do is map these curves to each other. But maybe next time we will prove both of these results. So we will look at this closely, slowly so you have a bit more time to make sure that you assimilate and digest this construction because this is in some sense the most sophisticated version of these fundamental domain constructions. But the principle is exactly the same. So if you make sure that you really have understood it in the case of one-dimensional linear maps, this is just adding a little bit of geometric, little bit more geometric complexity, but the argument is exactly the same. So it should not present many difficulties. If you're struggling, it's probably because you haven't really looked at the one-dimensional case closely.


Hey learners, it is summer officially and I cannot believe that my son is done with kindergarten That just went so fast and slow all at the same time. So today in honor of summer. I wanted to get that doesn't work. I Wanted to give you our bucket list of 20 things that we're gonna do this summer. Thank you If you are new to Lauren learns, don't forget to subscribe Hit the like on this. I can't Hit like on this video if this is helpful to you and let's get started like I Turn homemade smoothies into homemade popsicles Dance a garden and if you can't do that Just find a little pot and a few seeds and do a fun experiment with your kids go to a farmer's market Eat a snow cone Picnic at the park visit your local library for lots of fun programs find a pool Find some water Get wet treat Tuesdays. This is something we're gonna try new this year But every Tuesday we are gonna find a treat go to the skating rink go to the movies We have lots of parks close by so we're gonna ride our bikes to the park go camping and make s'mores And if you can't go to the woods to go camping just camp in your backyard Go to a nearby splash pad every year. We go to the Renaissance festival I can't wait for that one enjoy some root beer floats outside run in the sprinkler Find a new park you've never been to before what we'd like to do is go on Google Just put in park close to where we are Scan around and click on one and that's where we end up.  We plan to make a lemonade stand this year and instead of trying to make some money We're gonna go ahead and give that lemonade away for free and bless our neighbors I think one of my favorites will be this year is making a summer video of memories I can't wait to see all the fun things my kids and I get to do this year I have such good memories With my brother and my mom when we were kids and I can't want to do the same with my kids So summer is a really special time around here. We try to like find lots of free fun things to do It doesn't have to cost a lot of money for you to create your own bucket list at home Just get creative with what you have around you Find some friends to share it with invite them to come along and I promise you're gonna have a great summer And let's go get messy have fun and learn something this summer. See you next week You


What's up guys, Mike over with Jaspy's doing a quick recap for the 2023 Tops Chrome Mixer number two. It was a random team done on fanatics live option style The break consisted of 10 blasters two hobby boxes and two jumbo boxes so Just a quick recap here. Let's get into it some numbered cards rookies refractors Shoes There will be some Acunias in here Later Brett baby pink Pink or a green Walsh out of 99 Get into some autos Mark of the Entos base auto Tyler Freeman out of 499. Here's a cool one Gabriel Moreno image variation rookie orange speckle out of 25 for the Diamondbacks very nice Keep it going carry carpenter refractor auto Rosario base auto drew waters gold wave auto out of 50 Cody Clemens refractor auto and some more. There's a nice one Josh Jung nice color match Josh young Sorry, I don't 150 Caleb Killian Adley Ruchman Prism refractor pink Acunia.  Thank you guys. Thank you to everyone who got into the break Thank you guys for watching and I'll see you guys in the next one. Thanks


Hi, I'm Zor. Welcome to Unisor Education. I would like to talk about statistical approach to correlation between different random variables. Well, this is part of the Unisor Advanced Course of Mathematics. It's presented on this website. I do suggest you to watch this lecture from the website because it has notes, very detailed notes, which can be actually considered as an independent textbook. So you have a textbook and a lecture because it's kind of like a real school or university or whatever. Anyway, we will talk about statistical correlation today. Well, first of all, we have to realize that everything related to statistics is based on theory of probabilities. Now, in this particular case, there are a few lectures in theory of probabilities part of this course dedicated to correlation coefficient, covariance and then correlation were introduced. I do suggest you to refresh your memory and go through these couple of lectures. And there are some nice examples, actually, which will basically give you the theoretical foundation for statistics, which I will be talking about today. Alright, so being as it may, let me just remind you very briefly the definition of covariance and correlation from theory of probabilities standpoint. So let's consider you have two random variables. Now, first of all, I have introduced the covariance between them, which is expectation of their product minus product of their expectation. Now, immediately, you see that if these two variables are independent from each other, then the expectation of the product is equal to product of their expectations and the whole covariance will be equal to zero. So in a way, covariance is kind of a measure of independence, if you wish. So if the variables are independent, then covariance is equal to zero. So the reverse is not actually always true, it might or might not be. So if covariance is equal to zero, it does not really mean that they are necessarily independent, but it might be or might not. Now, the second thing which I could introduce in theory of probabilities course was correlation coefficient between them, which is defined as covariance divided by square root of their variances, product of their variances. Now, what's good about this? Why did we scale down or up or whatever the covariance? For one very, very simple reason. Now, the correlation coefficient is still zero if these guys are independent of each other, because the numerator will be equal to zero. Now, for ultimately dependent on each other random variables, now what do I consider to be ultimately dependent? Well, this, if a is some constant, they are proportional to each other or equal to each other, one of the particular case when a is equal to one. So that's the ultimate dependency. One value completely determines another, no some accidents, no randomness or anything else. So this variable completely defines this one. What happens in this case? Well, in this case, I can state that r is equal to r between two of them, correlation coefficient is equal to plus or minus one, depending on the sign of this constant. So this constant is positive, which means that this random variable is increasing when this is increasing and decreasing when this is decreasing. The correlation is one and if the a is negative, which means they're going in the opposite direction, one up and another down or vice versa, then the coefficient will be minus one. And it can be proven that for any random pair of random variables, this particular correlation coefficient will always be between minus one and one. With minus one and one on both ends representing ultimate dependency and zero representing independence. Okay, so that's all about theoretical foundation of correlation. Now let's go back to statistical correlation. Now, what is statistical correlation? Well, if I do not have complete distribution and I cannot calculate all these things, instead I have, excuse me, sample values. So I have certain observations where I observed c and eta. Based on these observations, I can actually estimate, I can evaluate this particular coefficient, this correlation coefficient and then make a judgment based on the value which I have obtained. How these two random variables are dependent on each other or not. So let's talk about sampling. Now, if I want to calculate something like this or like this, now obviously, if I have a sample of c, I can calculate this, I can calculate, if I have a sample of eta, I can calculate that and I can calculate separately their mathematical expectations. How about this one? Well, this is a completely different story because this means that they are together arranged into some kind of a calculation. In this case, it's a multiplication. Now, what does it mean? It means that I have to know the mutual distribution of these two variables, which means what's the probability of this taking one value and this taking some other value. If I have, at the same time, so if I have this mutual distribution, then I can actually calculate this expectation of their product. Now, since we are talking about statistics, nobody gives us any distribution beforehand. All we need is just sampling. But that's what's very important in this case. For instance, c is observed and it takes value. In any experiments, it took these values. Now, eta took these values. What's important is to synchronize these two experiments, which means whenever c took value x1 at the same time or at the same condition or under the same circumstances, eta took value y1. So, we need combined observation. That's what's very important. So, we observe and we see that c took x1 and eta took y1. Then we observe another time and then it will be x2 and y2. And then we observe the nth time, we have xn and yn. So, the observation should be mutual to be able to calculate this. This is very important. I cannot measure separately, have n experiments with c and get n values and then completely unrelated to this event take some experiments with eta and take these values. These are not related to each other, so we cannot really combine them into mutual distribution, which is required for this. Only if we can evaluate the probability, the frequency, whatever it is on their simultaneous taking certain values. So, that's why these n experiments should be experiments with both time or under the same circumstances. Now, what can it be? Just for example, it can be, let's say in the financial market, it can be, for instance, the price, closing price of some stock and at the same time you have something like index like Nasdaq 100 index observed at exactly the same moment at the price closing at the end of the day, let's say. So, you have, for instance, International Business Machine Corporation, IBM and you have Nasdaq 100. So, the day has closed and at the closing bell you have the value of one and you have the value of another. Then you have the next day, do exactly the same thing and the next and the next. So, if you observe n times in a row and each time you have both values at the same time, then you can actually talk about some statistical calculations, et cetera. Right now, if we have exactly this type of a situation, so when we can relate x1 to y1, x2 to y2 and xn to x to yn, then we can talk about calculation of their average of their product, mean product. Okay, here it is. So, we have xc taking the value xk and eta taking the value yk and I intentionally use the same index k in both cases because it's supposed to be, observations supposed to give both values, value of xc and value of eta at the same time or under the same circumstances. Now, if this is given to us and we have n observations, then we can definitely start calculating. What's the mean value, statistical mean value of c? I have to have, I have to summarize all the values from one to n and divide it by n, right? Now, similarly, I can calculate this divided by n. Similarly, I can calculate variance of xc, which is, it's a average of the square of deviation from the mean, right? So, deviation from the mean is xk minus e of xc. Square and we have to divide it by number of experiments and usually we put n minus one, I hope you remember that, to have unbiased variation. Well, with large n it doesn't really matter, but with a smaller n there is some noticeable difference. So, this is a variation of the variance based on statistical characteristics we have received based on sample where e of xc is this one. And similarly, obviously we have 4 eta, we have deviation from eta square, summarized and divided by n minus one. Great. So, all which is remaining is this one. What is the mean value of their product? Well, that's very easy. Since we know that c times eta took values xk times yk at each experiment, we have n experiments, the value of this new random variable will be observed as xk, yk on the case experiment. So, I have to add them together and divide it by n and that's my mean value. So, and this is c times eta. So, we have evaluated each component of this. So, we have evaluated this, then we can multiply these two together and then we can divide it by this. Now, the correlation coefficient is actually calculated. Okay, fine. So, we've done that. Now, we have to make certain... I mean, why do we do it? But we did it just because we would like to make some kind of a judgment on whether these two random variables are or are not related or dependent on each other, etc. Well, obviously since we know that our r is from minus one to one with value on the edges of this interval basically representing almost like linear dependency and value in the middle, which is zero, almost represent in dependency. So, therefore, based on this particular value which we have obtained using these calculations, we can make certain judgment just basically positioning our correlation coefficient on this interval from minus one to one. Now, obviously if it's very, very close to zero then you can say that, you know, what probably there is no dependency. If it's very, very close to one or to minus one you can say that probably dependency is relatively strong. So, what are the guidelines? So, whatever I'm saying right now is rather subjective. And it definitely depends on concrete circumstances. But traditionally and generally speaking, etc., whatever you want to say about this if the absolute value of r is less than 0.1 so it means from minus 0.1 to plus 0.1 we probably should consider that there is no correlation in this case. If it's from 0.1 to, let's say, 0.4 I would say there is a weak correlation. If it's from 0.4 to 0.7 moderate correlation and if it's above 0.7 then it's probably strong correlation. And again, the closer this is absolute value to 1 the stronger the correlation. And the closer it is to 0, the weaker is correlation. Now, these are no more than some kind of a guidelines and probably rather subjective, my own personal opinion would be and it's definitely not like universally acceptable kind of classification. So, most likely in different practical cases you will have different opinions about this. So, that's how we classify the correlation. Now, let's consider that we have established that there is some kind of a dependency let's say strong dependency, strong correlation between various of two different random variables. The next question is, can I say that one random variable influenced another or caused another to be such and such, to take such and such value? So, the cause and effect, causality, as we are saying, how is it related to correlation? Well, that's not such an easy question. Generally speaking, correlation doesn't mean causality. Because, well, here is what? Let's consider you have two random variables, c and eta and their coefficient of correlation is 0.9 which signifies a rather strong correlation. Now, is it c causing eta to take value which is like in sync with c or is it eta which is causing c to take this particular value? Or maybe there is some kind of a new random variable which we don't even know about and it caused both c and eta to take relatively synchronous value. We don't know that. I mean, it's not in the mathematics, let's put it this way. We might know about this from some other subjects like chemistry, physics, etc. geology, medicine, whatever you want, but it's not really in the mathematics. So, whenever you have calculated the correlation between two different random variables based on their statistical behavior, it's not really up to you basically to say that one is the cause for another without any kind of additional consideration. But here is some consideration which might actually be important. I was talking that both c and eta are supposed to be observed under the same circumstances or at the same time. Well, what if it's the same circumstance? For instance, we are experimenting with some drug and effect of that drug on the same person. In this case, the same person is something which basically combines, unifies these two things, the drug and the effect of this drug. So, if we always have this kind of a dependency between how much drug we get and how much effect we have from this drug, then probably there is a cause which is drug and effect. But it's not in the mathematics, it's completely from the different areas from the medicine. Similarly, in financial industry, for instance, you are measuring how much money the Federal Reserve Bank pumped into the market at, let's say, 2 o'clock in the afternoon. And what is the closing price of the market, some index at 4 o'clock in the afternoon? Well, there is actually some kind of a dependency here and obviously something which happens first might be the cause for something which happens second. Definitely not the other way around. So, at least you might have some judgment about cause and effect in some cases. But again, let me repeat that generally speaking, the mathematics alone, just the value of the correlation doesn't really give you any other information, but just put this number and there is some dependency, strong or weak or moderate or no dependency at all. It does not provide information about causality. So, correlation and causality are different things and causality can be derived from outside of the mathematical areas. And that's basically the end of this lecture. I will probably have some examples of correlation between different random variables and some future lectures. Thanks very much and good luck.