Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review
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paper_title: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

paper_content:

Gibbs Measures.- General Thermodynamic Formalism.- Axiom a Diffeomorphisms.- Ergodic Theory of Axiom a Diffeomorphisms.
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paper_title: Livsic theorems for connected Lie groups

paper_content:

Let $\phi$ be a hyperbolic diffeomorphism on a basic set $\Lambda$ and let $G$ be a connected Lie group. Let $f : \Lambda \rightarrow G$ be Holder. Assuming that $f$ satisfies a natural partial hyperbolicity assumption, we show that if $u : \Lambda \rightarrow G$ is a measurable solution to $f=u\phi \cdot u^{-1}$ a.e., then $u$ must in fact be Holder. Under an additional centre bunching condition on $f$, we show that if $f$ assigns `weight' equal to the identity to each periodic orbit of $\phi$, then $f = u\phi \cdot u^{-1}$ for some Holder $u$. These results extend well-known theorems due to Livsic when $G$ is compact or abelian.
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paper_title: On Devaney's Definition of Chaos

paper_content:

Chaotic dynamical systems have received a great deal of attention in recent years (see for instance [2], [3]). Although there has been no universally accepted mathematical definition of chaos, the popular text by Devaney [1] isolates three components as being the essential features of chaos. They are formulated for a continuous map f: X -> X on some metric space X (to avoid degenerate cases we will assume in this note that X is not a finite set). The first of Devaney's three conditions is that f is transitive; that is, for all non-empty open subsets U and V of X there exists a natural number k such that f k(U) n V is nonempty. In a certain sense, transitivity is an irreducibility condition. The second of Devaney's conditions is that the periodic points of f form a dense subset of X. Devaney refers to this condition as an "element of regularity" ([1], p. 50). The final condition is called sensitive dependence on initial conditions; f verifies this property if there is a positive real number 8 (a sensitivity constant) such that for every point x in X and every neighborhood N of x there exists a point y in N and a nonnegative integer n such that the nth iterates fW(x) and ff(y) of x and y respectively, are more than distance 8 apart. This sensitivity condition captures the idea that in chaotic systems minute errors in experimental readings eventually lead to large scale divergence. Sensitive dependence on initial conditions is thus widely understood as being the central idea in chaos.
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paper_title: Geodesic flows on closed Riemann manifolds with negative curvature

paper_content:

Free-radical graft polymerization of an unsaturated monomer like acrylic acid, methacrylic acid and so forth on mineral oils such as rubber processing and/or extending oils provides a "grafted oil" which can be used to extend polymers such as carboxylated SBR latices with substantial retention of their original adhesive properties. These grafted modified oils, also, can be used as seeds or emulsifiers, when emulsified, in emulsion polymerization. These grafted oils can be used as plasticizers for plastics and dry rubbers.
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paper_title: Stable Ergodicity of Certain Linear Automorphisms of the Torus,toappear

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We find a class of ergodic linear automorphisms of TN that are stably ergodic. This class includes all non-Anosov ergodic automorphisms when N = 4. As a corollary, we obtain the fact that all ergodic linear automorphism of TN are stably ergodic when N iU 5.
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paper_title: PARTIALLY HYPERBOLIC DYNAMICAL SYSTEMS

paper_content:

Smooth dynamical systems having contracting and expanding invariant foliations (of not necessarily complementary dimensions) are investigated. Ergodicity and the K-property are established for such dynamical systems under additional assumptions.
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paper_title: Stable ergodicity and julienne quasi-conformality

paper_content:

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.
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paper_title: Stably ergodic diffeomorphisms

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An examination of the human heart is carried out by means of ultrasound waves emitted by piezo-electric crystals. The crystals are inserted in a holder adapted to be placed on the body of a patient; they are arranged in a row and have parallel axes of radiation. The crystals are cyclically excited at a high scanning frequency, and the echo signals reflected by the heart walls and received by the crystals are displayed on the screen of a cathode ray tube to obtain a cross-sectional picture of the heart. In a modification of the invention, the holder comprises two or more rows of crystals radiating at different angles; in this case the echo signals related to a common point of the heart wall are superposed on the screen of the cathode ray tube by means of a coordinate transformation.
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paper_title: Topological chaos: what may this mean ?

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We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others.
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paper_title: Analysis On Lie Groups With Polynomial Growth

paper_content:

Thank you for reading analysis on lie groups with polynomial growth. Maybe you have knowledge that, people have search hundreds times for their favorite readings like this analysis on lie groups with polynomial growth, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they are facing with some infectious virus inside their desktop computer. analysis on lie groups with polynomial growth is available in our digital library an online access to it is set as public so you can get it instantly. Our digital library hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the analysis on lie groups with polynomial growth is universally compatible with any devices to read.
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paper_title: A note about topologically transitive cylindrical cascades

paper_content:

Letσ: Σ → Σ be a topologically mixing shift of finite type. Letβ: Σ → ℝ be a continuous function, and σ β be the skew-product ofσ byβ. Assume that σ β has a positive semi-orbit that reaches any upper height, and any lower height. Then, arbitrarilyC 0-close toβ there exists a Holder mapβ′: Σ → ℝ such that the skew-product\(\sigma _{\beta ^\prime } \) ofσ byβ′ is topologically transitive.
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paper_title: Stable Ergodicity for Smooth Compact Lie Group Extensions of Hyperbolic Basic Sets

paper_content:

We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C 2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the C r -topology for all r> 0( herer need not be an integer and C 1 is replaced by Lipschitz). Moreover, when r ≥ 2, we show that there is a C 2 -open and C r -dense subset of C r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R m -extensions, thereby generalizing a result of Niticu aa nd Pollicott, and on stable mixing for suspension flows.
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paper_title: Examples of topologically transitive skew-products

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Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group, ::: and $\beta:\Sigma\to G$ is a continuous function, ::: denote by $\sigma_\beta$ the skew-product of $\sigma$ by ::: $\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ ::: for which the skew-products $\sigma_\beta$ are topologically ::: transitive for sets of parameters of full measure. If $G$ is ::: a connected semisimple matrix Lie group, we show examples of functions $\beta$ for which the ::: skew-products $\sigma_beta$ are topologically transitive.
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paper_title: Stable topological transitivity properties of ℝ n -extensions of hyperbolic transformations

paper_content:

We consider ℝ n skew-products of a class of hyperbolic dynamical systems. It was proved by Niţică and Pollicott [Transitivity of Euclidean extensions of Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 25 (2005), 257–269] that for an Anosov diffeomorphism ϕ of an infranilmanifold Λ there is (subject to avoiding natural obstructions) an open and dense set f :Λ→ℝ N for which the skew-product ϕ f ( x , v )=(ϕ( x ), v + f ( x )) on Λ×ℝ N has a dense orbit. We prove a similar result in the context of an Axiom A hyperbolic flow on an attractor.
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paper_title: A NOTE ABOUT STABLE TRANSITIVITY OF NONCOMPACT EXTENSIONS OF HYPERBOLIC SYSTEMS

paper_content:

Let $f:X\to X$ be the restriction to a ::: hyperbolic basic set of a smooth diffeomorphism. If $G$ is the ::: special Euclidean group ::: $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there ::: exists an open and dense subset of stably transitive transformations. ::: If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, ::: we show that an open and dense set of $C^2$ $G$-extensions satisfying a ::: certain separation condition are transitive. The separation condition is necessary.
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paper_title: TRANSITIVITY OF EUCLIDEAN-TYPE EXTENSIONS OF HYPERBOLIC SYSTEMS

paper_content:

Let f:X→X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of Cr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values of n≥3. More generally, we consider Euclidean-type groups Gn where G is a compact connected Lie group acting linearly on n. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.
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paper_title: Stable transitivity of Euclidean group extensions

paper_content:

The topological transitivity of non-compact group extensions of topologically mixing subshifts of finite type has been studied recently by Niticua. We build on these methods, and give the first examples of stably transitive non-compact group extensions of hyperbolic dynamical systems. Our examples include extensions of hyperbolic basic sets by the Euclidean group SE(n) for n even, n ≥ 4.
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paper_title: Transitivity of Heisenberg group extensions of hyperbolic systems

paper_content:

We consider skew-extensions with fibre the standard real Heisenberg group of a uniformly hyperbolic dynamical system.We show that among the Cr extensions (r > 0) that avoid an obvious obstruction, those that are topologically transitive contain an open and dense set. More precisely, we show that an -extension is transitive if and only if the -extension given by the Abelianization of is transitive.A new technical tool introduced in the paper, which is of independent interest, is a diophantine approximation result. We show, under general conditions, the existence of an infinite set of approximate positive integer solutions for a diophantine system of equations consisting of a quadratic indefinite form and several linear equations. The set of approximate solutions can be chosen to point in a certain direction. The direction can be chosen from a residual subset of full measure of the set of real directions solving the system of equations exactly.Another contribution of the paper, which is used in the proof of the main result, but it is also of independent interest, is the solution of the so-called semigroup problem for the Heisenberg group. We show that for a subset , which avoids any maximal semigroup with non-empty interior, the closure of the semigroup generated by S is actually a group.
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paper_title: Stable Ergodicity for Smooth Compact Lie Group Extensions of Hyperbolic Basic Sets

paper_content:

We obtain sharp results for the genericity and stability of transitivity, ergodicity and mixing for compact connected Lie group extensions over a hyperbolic basic set of a C 2 diffeomorphism. In contrast to previous work, our results hold for general hyperbolic basic sets and are valid in the C r -topology for all r> 0( herer need not be an integer and C 1 is replaced by Lipschitz). Moreover, when r ≥ 2, we show that there is a C 2 -open and C r -dense subset of C r -extensions that are ergodic. We obtain similar results on stable transitivity for (non-compact) R m -extensions, thereby generalizing a result of Niticu aa nd Pollicott, and on stable mixing for suspension flows.
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paper_title: Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups

paper_content:

Assume that $X$ is a hyperbolic basic set for $f:X\to X$. ::: We show new examples of Lie group fibers $G$ for which, in the class of $C^r, r>0,$ $G$-extensions ::: of $f$, those that are transitive are open and dense. The fibers are semidirect ::: products of compact and nilpotent groups.
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paper_title: A NOTE ABOUT STABLE TRANSITIVITY OF NONCOMPACT EXTENSIONS OF HYPERBOLIC SYSTEMS

paper_content:

Let $f:X\to X$ be the restriction to a ::: hyperbolic basic set of a smooth diffeomorphism. If $G$ is the ::: special Euclidean group ::: $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there ::: exists an open and dense subset of stably transitive transformations. ::: If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, ::: we show that an open and dense set of $C^2$ $G$-extensions satisfying a ::: certain separation condition are transitive. The separation condition is necessary.
---
paper_title: A NOTE ABOUT STABLE TRANSITIVITY OF NONCOMPACT EXTENSIONS OF HYPERBOLIC SYSTEMS

paper_content:

Let $f:X\to X$ be the restriction to a ::: hyperbolic basic set of a smooth diffeomorphism. If $G$ is the ::: special Euclidean group ::: $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there ::: exists an open and dense subset of stably transitive transformations. ::: If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, ::: we show that an open and dense set of $C^2$ $G$-extensions satisfying a ::: certain separation condition are transitive. The separation condition is necessary.
---
paper_title: TRANSITIVITY OF EUCLIDEAN-TYPE EXTENSIONS OF HYPERBOLIC SYSTEMS

paper_content:

Let f:X→X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of Cr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values of n≥3. More generally, we consider Euclidean-type groups Gn where G is a compact connected Lie group acting linearly on n. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.
---
paper_title: Transitivity of Heisenberg group extensions of hyperbolic systems

paper_content:

We consider skew-extensions with fibre the standard real Heisenberg group of a uniformly hyperbolic dynamical system.We show that among the Cr extensions (r > 0) that avoid an obvious obstruction, those that are topologically transitive contain an open and dense set. More precisely, we show that an -extension is transitive if and only if the -extension given by the Abelianization of is transitive.A new technical tool introduced in the paper, which is of independent interest, is a diophantine approximation result. We show, under general conditions, the existence of an infinite set of approximate positive integer solutions for a diophantine system of equations consisting of a quadratic indefinite form and several linear equations. The set of approximate solutions can be chosen to point in a certain direction. The direction can be chosen from a residual subset of full measure of the set of real directions solving the system of equations exactly.Another contribution of the paper, which is used in the proof of the main result, but it is also of independent interest, is the solution of the so-called semigroup problem for the Heisenberg group. We show that for a subset , which avoids any maximal semigroup with non-empty interior, the closure of the semigroup generated by S is actually a group.
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Title: Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

Section 1: Introduction
Description 1: Provide an overview of dynamical systems, topological transitivity, and related concepts such as topological chaos, ergodicity, and hyperbolic systems.

Section 2: Statements of Available Results
Description 2: Summarize the outcomes of various classes of Lie groups concerning the main conjecture about topological transitivity among Hölder cocycles with subexponential growth.

Section 3: Γ Compact Connected Lie Group
Description 3: Discuss results related to extensions where the fiber Γ is a compact connected Lie group, including the proof of Brin and other improvements and generalizations.

Section 4: Γ = R^n
Description 4: Explore the results concerning extensions with fiber R^n, including the equivalence conditions for transitivity and the significance of periodic data.

Section 5: Γ Is a Euclidean-Type Group
Description 5: Detail findings for extensions where Γ is an Euclidean-type group, such as SE(n), including specific challenges for odd and even n.

Section 6: Γ Is a Nilpotent Lie Group
Description 6: Present results for extensions where Γ is a nilpotent Lie group, highlighting related diophantine approximation results.

Section 7: Γ Is a Compact and Nilpotent Semidirect Product
Description 7: Discuss the outcomes for extensions where Γ is a compact and nilpotent semidirect product, including the criteria for a group to be classified as such.

Section 8: Γ Is a Non-Compact Semisimple Lie Group
Description 8: Describe open problems and partial results concerning non-compact semisimple Lie groups, such as SL(2, R) and Sp(n, R).

Section 9: Criterion for Transitivity
Description 9: Introduce criteria used to establish topological transitivity, such as accessibility and density of recurrent points, including Theorem 11.

Section 10: Admissible Sequences of Products of Holonomies
Description 10: Explain the method for obtaining elements in the set L_β using periodic heteroclinic cycles and approximations.

Section 11: Semigroup Problem
Description 11: Discuss the semigroup problem for various Lie groups, aiming to prove that certain sets generate the whole group as a semigroup.