Exponential Decay of Lebesgue Numbers

TL;DR

This paper investigates how the Lebesgue numbers of open covers in topological dynamical systems decay exponentially, establishing a bound involving topological entropy and dimension, with various corollaries and examples.

Contribution

It introduces a novel relationship between the exponential decay rate of Lebesgue numbers and topological entropy in dynamical systems.

Findings

Topological entropy is bounded by the decay rate times the dimension.

The paper provides new bounds and relationships involving Lebesgue numbers and entropy.

Several examples illustrate the theoretical results.

Abstract

We study the exponential rate of decay of Lebesgue numbers of open covers in topological dynamical systems. We show that topological entropy is bounded by this rate multiplied by dimension. Some corollaries and examples are discussed.