The aperiodic complexities and connections to dimensions and Diophantine approximation

TL;DR

This paper introduces new measures of aperiodic complexity in dynamical systems, relates them to geometric and topological properties, and connects these concepts to Diophantine approximation and periodic orbit distribution.

Contribution

It defines the aperiodic complexities $\\cal{F}$ and $\cal{G}$, establishes their relations to system topology and geometry, and links aperiodic orbits to Diophantine approximation properties.

Findings

Relations between aperiodic complexities and topological entropy

Classification of orbits via Diophantine approximation constants

A metric version of the closing lemma for CAT(-1) spaces

Abstract

In their earlier work (Ergodic Th. Dynam. Sys., 34: 1699 -1723, 10 2014), the authors introduced the so called F-aperiodic orbits of a dynamical system on a compact metric space X, which satisfy a quantitative condition measuring its recurrence and aperiodicity. Using this condition we introduce two new quantities $\cal{F}$, $\cal{G}$, called the `aperiodic complexities', of the system and establish relations between $\cal{F}$, $\cal{G}$ with the topology and geometry of X. We compare them to well-know complexities such as the box-dimension and the topological entropy. Moreover, we connect our condition to the distribution of periodic orbits and we can classify an F-aperiodic orbit of a point x in X in terms of the collection of the introduced approximation constants of x. Finally, we discuss our results for several examples, in particular for the geodesic flow on hyperbolic manifolds. For each of our examples there is a suitable model of Diophantine approximation and we classify F-aperiodic orbits in terms of Diophantine properties of the point x. As a byproduct, we prove a `metric version' of the closing lemma in the context of CAT(-1) spaces.