On INVARIANT Ε-Scrambled SETS

Abstract

This article is devoted to the study of invariant ε-scrambled sets. We show that every topologically mixing map with at least one fixed point contains at least one such set. Additionally we show that this condition can be weakened in the case of symbolic dynamics, e.g. mixing can be replaced by transitivity. Some relations between mixing and proximal relation are also studied.

Invariant scrambled sets, uniform rigidity and weak mixing

“…As a byproduct of Theorem C, the above mentioned conjecture from [6] has an affirmative answer, because in Theorem A we construct a weakly mixing, proximal and uniformly rigid system with a fixed point but without invariant δ -scrambled set.…”

Recent development of chaos theory in topological dynamics

“…Combining Theorems 3.14 and 3.19, the above mentioned conjecture from [9] has an affirmative answer. This is because by Theorem 3.14 there exist weakly mixing, proximal and uniformly rigid systems which have a fixed point, but by Theorem 3.19 they do not have any invariant δ -scrambled set.…”

“…They also conjectured in [9] that there exists a weakly mixing system which has a fixed point but without invariant δ -scrambled sets. The authors in [24] found that the existence of invariant δ -scrambled sets is relative to the property of uniform rigidity.…”

A dynamical version of the Kuratowski–Mycielski theorem and invariant chaotic sets

“…In [25] Yuan and Lü showed that if a transitive system has a fixed point then it has a dense, σ-Cantor, invariant scrambled set. In [3] Balibrea, Guirao and Oprocha showed that if a strongly mixing system has a fixed point then it has a dense, σ-Cantor, invariant δ-scrambled set for some δ > 0. In [11] Foryś, Oprocha and Wilczyński showed that if a compact dynamical system has the specification property and has a fixed point then it has a dense, σ-Cantor, invariant distributionally δ-scrambled set for some δ > 0.…”

“…As f n (U 1 ) = X for all n ≥ N, there exists a point z ∈ U 1 such that f k 2 (z) = y. Note that f k 3…”