Li-Yorke pairs of full Hausdorff dimension for some chaotic dynamical systems

Abstract

pairs of full Hausdorff dimension for some chaotic dynamical systems

Chaotic sets and Hausdorff dimension for Lüroth expansions

“…In the same setting, F. Blanchard et al in [4] shown that if the topological entropy of (X, f ) is positive, then (X, f ) is Li-Yorke chaotic. The dimensional properties of the sets given in Definition 1.1 have received attention recently -see [22]. 1.2.…”

Chaotic and topological properties of continued fractions

“…The sizes of the scrambled sets have been considered for many dynamical systems from the sense of the measure, dimension, and topology, see also [2], [16], [17], [22], [23] etc. In 1995, Xiong [25] proved that there exists a scrambled set of {1, 2, • • • , N } N of full Hausdorff dimension .…”

Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems

“…Li-Yorke pairs are the pairs of points that approach each other for some sequence of moments in the time evolution and that remain separated for other sequences of moments. In [5] was discussed on the Hausdorff dimension of the set of Li-Yorke pairs for some simple classical chaotic dynamical systems. It showed that, if the dynamical system in its invariant set is topologically conjugate to the full shift symbolic dynamical system and its invariant set is self-similar or a product of self-similar sets, then its Li-Yorke pairs have full Hausdorff dimension in the invariant set.…”

“…It showed that, if the dynamical system in its invariant set is topologically conjugate to the full shift symbolic dynamical system and its invariant set is self-similar or a product of self-similar sets, then its Li-Yorke pairs have full Hausdorff dimension in the invariant set. This result can be applied to simple classical models of "chaotic" dynamics like the tent map, the Bakers transformation, Smales horseshoe, and solenoid-like systems (see [5]) since these kinds of systems have invariant sets of self-similar or a product of self-similar sets in which the systems are topologically conjugate to full shift. To prove that Li-Yorke pairs have full dimension for more general hyperbolic systems could be a task for further research [5], which is one topic we are going to study in this paper.…”

“…We prove that Li-Yorke pairs of A-coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Further more we count Hausdorff dimension of "chaotic invariant set" for some kinds of A-coupled-expanding maps.…”