Levels of generalized expansiveness

TL;DR

This paper explores various levels of generalized expansiveness in dynamical systems, providing new characterizations and examples, including systems with arbitrary countable van der Waerden depth, thus addressing open questions.

Contribution

It introduces a classification of generalized expansive systems, characterizes n-expansiveness for countable systems, and constructs examples with specified van der Waerden depths.

Findings

Characterization of n-expansiveness for countable systems

Construction of systems with arbitrary countable van der Waerden depth

Solutions to open questions in the literature

Abstract

We study a class of generalized expansive dynamical systems for which at most countable orbits can be accompanied by an arbitrary given orbit. Examples of different levels of generalized expansiveness are constructed. When the dynamical system is countable, a characterization of n-expansiveness is given for any natural number n, and as a consequence examples of dynamical systems with van der Waerden depth equal to any given countable ordinal are demonstrated, which solves open questions existing in the literature.