Measures related to (e,n)-complexity functions

TL;DR

This paper investigates the properties of (e,n)-complexity functions in dynamical systems, focusing on the measures derived from separated sets and their invariance under subexponential growth conditions.

Contribution

It introduces a detailed analysis of measures associated with (e,n)-complexity functions and proves their invariance when the complexity grows subexponentially.

Findings

Measures constructed from (e,n)-separated sets are invariant under subexponential growth.

The behavior of (e,n)-complexity functions is linked to properties of specific invariant measures.

The study connects complexity functions with topological entropy and separability in dynamical systems.

Abstract

The (e,n)-complexity functions describe total instability of trajectories in dynamical systems. They reflect an ability of trajectories going through a Borel set to diverge on the distance $\epsilon$ during the time interval n. Behavior of the (e, n)-complexity functions as n goes to infinity is reflected in the properties of special measures. These measures are constructed as limits of atomic measures supported at points of (e,n)-separated sets. We study such measures. In particular, we prove that they are invariant if the (e,n)-complexity function grows subexponentially. Keywords: Topological entropy, complexity functions, separability.