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Similarities and differences between specification and non-uniform specification

Part of: Topological dynamics Smooth dynamical systems: general theory

Published online by Cambridge University Press:  15 April 2024

WANSHAN LIN Open the ORCID record for WANSHAN LIN [Opens in a new window] ,
XUETING TIAN  and
CHENWEI YU
WANSHAN LIN*
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China (e-mail: xuetingtian@fudan.edu.cn, 23110180052@m.fudan.edu.cn)
XUETING TIAN
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China (e-mail: xuetingtian@fudan.edu.cn, 23110180052@m.fudan.edu.cn)
CHENWEI YU
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China (e-mail: xuetingtian@fudan.edu.cn, 23110180052@m.fudan.edu.cn)
*
e-mail: 21110180014@m.fudan.edu.cn
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Abstract

Pavlov [Adv. Math. 295 (2016), 250–270; Nonlinearity 32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Keywords

specificationinvariant measureperiodic orbitgeneric pointsubshift

MSC classification

Primary: 37B10: Symbolic dynamics
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37C25: Fixed points, periodic points, fixed-point index theory

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 12 , December 2024 , pp. 3501 - 3529
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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