Uniform tail entropy for real analytic maps

TL;DR

This paper establishes a uniform bound on the tail entropy of real analytic maps on compact manifolds, showing it diminishes to zero as the scale shrinks, which enhances understanding of their complexity at small scales.

Contribution

It proves the existence of a universal function bounding the tail entropy of all real analytic maps on compact manifolds, highlighting a uniform decay property.

Findings

Tail entropy is uniformly bounded by a function tending to zero as scale decreases.

The bound applies universally to all real analytic maps on the manifold.

This result provides a new quantitative measure of the complexity of real analytic dynamical systems.

Abstract

Let $M$ be a compact real analytic manifold of finite dimension. There is a function $a: (0,+\infty)\to [0,+\infty)$ with $\lim_{t\to0}a(t)=0$ such that, the tail entropy $h^{*}(f,\varepsilon)$ of any real analytic map $f$ on $M$ is uniformly bounded above by the scale $a(\varepsilon)$.