Dynamics and eigenvalues in dimension zero

TL;DR

This paper explores the relationship between the eigenvalues of the induced map on zero-dimensional Čech homology and the dynamical complexity of continuous maps on totally disconnected compact spaces, revealing that complex eigenvalues of modulus not equal to 0 or 1 indicate complicated dynamics.

Contribution

It establishes a novel connection between eigenvalues of the induced homology map and dynamical properties, contrasting with classical inequalities.

Findings

Eigenvalues of the induced map reflect dynamical complexity.

Complex eigenvalues with modulus not equal to 0 or 1 correspond to complicated dynamics.

Contrasts with classical Manning's inequality.

Abstract

Let $X$ be a compact, metric and totally disconnected space and let $f:X\to X$ be a continuos map. We relate the eigenvalues of $f_{*}:\check{H}_{0}(X;\mathbb{C})\to\check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0,1 is an eigenvalue. This stands in contrast with the classical Manning's inequality.