Transitive cylinder flows whose set of discrete points is of full Hausdorff dimension (original) (raw)

On rational ergodicity of certain cylinder flows

2011

We construct new examples of cylinder flows, given by skew product extensions of irrational rotations on the circle, that are ergodic, rationally ergodic and weakly homogeneous. Our examples exhibit gaussian distribution of the number of visits to zero.

Cyclicity of chaotic attractors in one-dimensional discontinuous maps

Mathematics and Computers in Simulation, 2014

A chaotic attractor may consist of some number of bands (disjoint connected subsets). In continuous maps multi-band chaotic attractors are cyclic, that means every generic trajectory visits the bands in the same order. We demonstrate that in discontinuous maps multi-band chaotic attractors may be acyclic. Additionally, a simple criterion is proposed which allows to distinguish easily between cyclic and acyclic chaotic attractors.

Invariant geometric properties of a class of 3D chaotic flows

Physica D-nonlinear Phenomena, 2000

This article extends the analysis developed by Giona and Adrover [M. Giona, A. Adrover, Phys. Rev. Lett. 81 (1998) 3864] for 2D area-preserving diffeomorphisms to 3D volume-preserving C ∞ -diffeomorphisms of the 3D torus topologically conjugate to a linear map. The article analyzes the invariant geometric properties of vector dynamics and surface element evolution in 3D systems and provides an analytic expression for the probability measure describing pointwise statistical properties of the unstable foliations in the hyperbolic case. The convergence properties of this measure are addressed starting from the dynamics of surface elements. The application of the methods developed to physically realizable 3D chaotic flows such as ABC flow is discussed in detail.

Composition of Chaotic Maps with an Invariant Measure

We generate new hierarchy of many-parameter family of maps of the interval [0, with an invariant measure, by composition of the chaotic maps of reference . Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of these maps analytically, where the results thus obtained have been approved with numerical simulation. In contrary to the usual one-dimensional maps and similar to the maps of reference [1], these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor at certain region of parameters values, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at these values of parameter whose Lyapunov characteristic exponent begins to be positive.

On the topology of solenoidal attractors of the cylinder

Annales de l'Institut Henri …, 2006

We study the dynamics of skew product endomorphisms acting on the cylinder R/Z × R, of the form (θ, t) → θ, λt + τ (θ) , where 2 is an integer, λ ∈ (0, 1) and τ : R/Z → R is a continuous function. We are interested in topological properties of the global attractor Ω λ,τ of this map. Given and a Lipschitz function τ , we show that the attractor set Ω λ,τ is homeomorphic to a closed topological annulus for all λ sufficiently close to 1. Moreover, we prove that Ω λ,τ is a Jordan curve for at most finitely many λ ∈ (0, 1). These results rely on a detailed study of iterated "cohomological" equations of the form τ = L λ 1 µ 1 , µ 1 = L λ 2 µ 2 ,. .. , where L λ µ = µ • m − λµ and m : R/Z → R/Z denotes the multiplication by map. We show the following finiteness result: each Lipschitz function τ can be written in a canonical way as, τ = L λ 1 • • • • • L λ m µ, where m 0, λ 1 ,. .. , λ m ∈ (0, 1] and the Lipschitz function µ satisfies µ = L λ ρ for every continuous function ρ and every λ ∈ (0, 1].

Topological entropy of Devaney chaotic maps

Topology and its Applications, 2003

The infimum respectively minimum of the topological entropies in different spaces are studied for maps which are transitive or chaotic in the sense of Devaney (i.e., transitive with dense periodic points). After a short survey of results explicitly or implicitly known in the literature for zero and onedimensional spaces the paper deals with chaotic maps in some higher-dimensional spaces. The key role is played by the result saying that a chaotic map f in a compact metric space X without isolated points can always be extended to a triangular (skew product) map F in X × [0, 1] in such a way that F is also chaotic and has the same topological entropy as f . Moreover, the sets X × {0} and X × {1} are F -invariant which enables to use the factorization and obtain in such a way dynamical systems in the cone and in the suspension over X or in the space X × S 1 . This has several consequences. Among others, the best lower bounds for the topological entropy of chaotic maps on disks, tori and spheres of any dimensions are proved to be zero.

Entropy and exact Devaney chaos on totally regular continua

Discrete and Continuous Dynamical Systems, 2013

We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called P-Lipschitz maps (where P is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum X contains a free arc which does not disconnect X or if X contains arbitrarily large generalized stars then X admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entropies of transitive/Devaney chaotic maps is indicated.