An invitation to rough dynamics: zipper maps

TL;DR

This paper introduces zipper maps, a class of irregular dynamical systems, and explores their complex behaviors including horseshoes, infinite entropy, and positive metric mean dimension, advancing understanding of irregular dynamics.

Contribution

It defines zipper maps and analyzes their dynamical properties, showing they exhibit high complexity such as horseshoes and positive metric mean dimension, which was previously unexplored.

Findings

Zipper maps admit horseshoes of all orders for many parameters.

They have infinite topological entropy.

They possess positive metric mean dimension.

Abstract

In the field of dynamical systems, it is not rare to meet irregular functions, which are typically H{\"o}lder but not Lipschitz (e.g. the Weierstrass functions). Our goal is to scratch the surface of the following question: what happens if we consider irregular maps and iterate them? We introduce the family of "zipper maps", which are irregular in the above sense, and study some of their dynamical properties. For a large set of parameters, the corresponding zipper map admits horseshoe of all orders; as an immediate consequence, every order on $k\ell$ points can be realized by $k$ orbits of length $\ell$ of the map.These maps have infinite topological entropy, and we refine this statement by showing that they have positive metric mean dimension with respect to the Euclidean metric, as well as by introducing other notions of higher complexity.Finally, we prove that every interval map (thus including zipper maps) have vanishing absolute metric mean dimension, proving a small case of the conjecture that the absolute metric mean dimension coincides with the topological mean dimension.