Local distributional chaos

TL;DR

This paper explores local distributional chaos in dynamical systems, showing that in symbolic spaces all points are DC1-points and that positive entropy implies many DC1-points, while in higher dimensions chaos can be less concentrated.

Contribution

It introduces the concept of DCi-points, proves that all points in symbolic spaces are DC1-points, and demonstrates that positive entropy leads to uncountably many DC1-points, with examples in higher dimensions.

Findings

All points in symbolic space are DC1-points.

Positive topological entropy implies uncountably many DC1-points.

Higher-dimensional systems can have positive entropy without DC2-points.

Abstract

In discrete dynamical system $(X, f)$ where $X$ is a topological space and $f \in C(X,X)$, three notions of distributional chaos were defined. They were denoted by $DC1, DC2$ and $DC3$. For interval systems such three notions coincide and they will be denoted by DC-chaos. Generally speaking we have $DC1 \subseteq DC2 \subseteq DC3$-chaos. We wonder if it is possible that chaos can concentrate in some points and develop a local idea of the distributional chaos. Answering to this question, in [12] is introduced the new notion of $DCi$-points for $i = 1,2,3$. Such special points are those in which DC-chaos of different types concentrate. Also in [12] it is proved that if $f$ is continuous interval map with positive topological entropy, then there is at least one DC1-point in the system. In this paper it is proved that in the symbolic space $(\Sigma, \sigma) $ where $\sigma$ is the shift map, every point of $\Sigma$ is a DC1-point. This result is necessary to prove one of the main results of the paper. If $h(f) > 0 $ for an interval system, then it has an uncountable set of DC1-points and moreover the set can be chosen perfect. In greater dimensions than one, we deal with triangular systems on $I^{2}$. In this case the relationship between topological entropy and different cases of distributional chaos is not clearly understood and several different results are possible. In the paper we use an example of $F$ given by Kolyada in [9] to prove that the corresponding two dimensional system $(I^{2}, F)$ has positive topological entropy but without containing DC2-points, proving that there is no concentration of DC2-chaos