Genericity of distributional chaos in non-autonomous dynamical systems

TL;DR

This paper investigates the nature of distributional chaos in non-autonomous discrete dynamical systems, revealing that certain chaos properties depend on the metric and are not generically present, with results varying between Cantor and interval systems.

Contribution

It resolves two open problems by showing the metric dependence of DC2 and the non-genericity of DC1 in non-autonomous systems, providing new insights into their chaotic behavior.

Findings

Limit function of pointwise convergent systems can be DC2 or not, depending on the metric.

DC1 chaos is not a generic property in convergent systems on the Cantor set.

DC1 chaotic systems are dense but not open in the space of convergent systems on the interval.

Abstract

In this paper we solve two open problems concerning distributional chaos in non-autonomous discrete dynamical systems stated in [4] and [17]. In the first problem it is wondered if the limit function of pointwise convergent non-autonomous system with positive topological entropy is DC2. We show that the answer to this problem depends on the given metric and can be both, positive or negative. In the second open problem it is wondered if to be DC1 is a generic property of pointwise convergent non-autonomous systems. We prove that the answer is negative for convergent systems on the Cantor set. Concerning interval systems, we show that DC1 chaotic systems form dense, but not open (nor closed) set in the space of non-autonomous convergent systems on the interval, independently of the metric we use.