Though the Arnold cat map has this property, the discrete cat map does not.
One of the two criteria for a system to have this property is that for any two neighborhoods U and V in the space, the iteration will eventually make the intersection of U and V non-negative. Though the Arnold cat map has this property, the discrete cat map does not. Devaney characterized this type of system as having dense periodic orbits, topological transitivity, and having the property that there exists a positive number d such that for any two points x and y, the orbits will eventually be (*) farther than d apart. This property is characterized by a positive Lyapunov exponent, and the logistic family has this property for a parameter between about 3.57 and 4. The Lorentz attractor and double pendulum display, for 10 points, what property of a dynamical system that is very sensitive to initial conditions?
One of the two criteria for a system to have this property is that for any two neighborhoods U and V in the space, the iteration will eventually make the intersection of U and V non-negative.
chaos