Expanding maps on sets which are almost invariant. Decay and chaos

On invariant sets topology

In this paper, we introduce and study a new topology related to a self mapping on a nonempty set.Let X be a nonempty set and let f be a self mapping on X. Then the set of all invariant subsets ofX related to f, i.e. f := fA X : f(A) Ag P(X) is a topology on X. Among other things,we nd the smallest open sets contains a point x 2 X. Moreover, we find the relations between fand To f . For insta...

Fluctuation Induced Almost Invariant Sets

Statistically optimal almost-invariant sets

Chaotic dynamical systems are often transitive, although this transitivity is sometimes very weak. It is of interest to divide the phase space into large regions, between which there is relatively little communication of trajectories. We present fast, simple algorithms to find such divisions. The present work builds on the results of Froyland and Dellnitz [G. Froyland, M. Dellnitz, Detecting an...

Amenable Actions and Almost Invariant Sets

In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets.

Decay of Correlations for Piecewise Expanding Maps

This paper investigates the decay of correlations in a large class of non-Markov one-dimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.