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Properties of invariant measures in dynamical systems with the shadowing property

Published online by Cambridge University Press:  14 March 2017

JIAN LI Open the ORCID record for JIAN LI [Opens in a new window]  and
PIOTR OPROCHA
JIAN LI
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, PR China email lijian09@mail.ustc.edu.cn
PIOTR OPROCHA
Affiliation:
AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Kraków, Poland National Supercomputing Centre IT4 Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy Modeling, 30. dubna 22, 70103 Ostrava, Czech Republic email oprocha@agh.edu.pl
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Abstract

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost one-to-one extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\unicode[STIX]{x1D700}>0$ the collection of ergodic measures (supported on almost one-to-one extensions of odometers) with entropy between $c$ and $c+\unicode[STIX]{x1D700}$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then, for every $c\geq 0$, ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 6 , September 2018 , pp. 2257 - 2294
Copyright
© Cambridge University Press, 2017 

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