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Dynamics and eigenvalues in dimension zero

Part of: Topological dynamics Low-dimensional dynamical systems Homology and cohomology theories

Published online by Cambridge University Press:  04 January 2019

LUIS HERNÁNDEZ-CORBATO Open the ORCID record for LUIS HERNÁNDEZ-CORBATO [Opens in a new window] ,
DAVID JESÚS NIEVES-RIVERA Open the ORCID record for DAVID JESÚS NIEVES-RIVERA [Opens in a new window] ,
FRANCISCO R. RUIZ DEL PORTAL Open the ORCID record for FRANCISCO R. RUIZ DEL PORTAL [Opens in a new window]  and
JAIME J. SÁNCHEZ-GABITES Open the ORCID record for JAIME J. SÁNCHEZ-GABITES [Opens in a new window]
LUIS HERNÁNDEZ-CORBATO
Affiliation:
Departamento de Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, 28031Madrid, Spain email luis.hcorbato@upm.es
DAVID JESÚS NIEVES-RIVERA
Affiliation:
Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040Madrid, Spain email davidjni@ucm.es, rrportal@ucm.es
FRANCISCO R. RUIZ DEL PORTAL
Affiliation:
Departamento de Álgebra, Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040Madrid, Spain email davidjni@ucm.es, rrportal@ucm.es
JAIME J. SÁNCHEZ-GABITES
Affiliation:
Departamento de Análisis Económico (Métodos cuantitativos), Facultad de Ciencias Económicas y Empresariales, Universidad Autónoma de Madrid, 28049Madrid, Spain email JaimeJ.Sanchez@uam.es
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Abstract

Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ 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 a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$.

Keywords

Čech homologyeigenvaluesadding machinesentropy

MSC classification

Primary: 37B40: Topological entropy 37E99: None of the above, but in this section
Secondary: 37B10: Symbolic dynamics 55N05: Čech types
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 9 , September 2020 , pp. 2434 - 2452
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Copyright
© Cambridge University Press, 2019

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