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Uniformly positive entropy of induced transformations

Part of: Classical measure theory Basic constructions Topological dynamics Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  28 December 2020

NILSON C. BERNARDES JR ,
UDAYAN B. DARJI Open the ORCID record for UDAYAN B. DARJI [Opens in a new window]  and
RÔMULO M. VERMERSCH
NILSON C. BERNARDES JR
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, RJ21945-970, Brazil (e-mail:ncbernardesjr@gmail.com)
UDAYAN B. DARJI*
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY40208-2772, USA
RÔMULO M. VERMERSCH
Affiliation:
Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis, SC88040-900, Brazil (e-mail:romulo.vermersch@gmail.com)
*
e-mail:ubdarj01@louisville.edu
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Abstract

Let $(X,T)$ be a topological dynamical system consisting of a compact metric space X and a continuous surjective map $T : X \to X$ . By using local entropy theory, we prove that $(X,T)$ has uniformly positive entropy if and only if so does the induced system $({\mathcal {M}}(X),\widetilde {T})$ on the space of Borel probability measures endowed with the weak* topology. This result can be seen as a version for the notion of uniformly positive entropy of the corresponding result for topological entropy due to Glasner and Weiss.

Keywords

continuous surjective mapsdynamicsprobability measuresProhorov metricweak* topology

MSC classification

Primary: 37B40: Topological entropy
Secondary: 28A33: Spaces of measures, convergence of measures 60B10: Convergence of probability measures 54B20: Hyperspaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 1 , January 2022 , pp. 9 - 18
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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