Miles, R and Ward, T orcid.org/0000-0002-8253-5767 (2015) Directional uniformities, periodic points, and entropy. Discrete and Continuous Dynamical Systems - Series B, 20 (10). pp. 3525-3545. ISSN 1531-3492
Abstract
Dynamical systems generated by d≥2 commuting homeomorphisms (topological Zd-actions) contain within them structures on many scales, and in particular contain many actions of Zk for 1≤k≤d. Familiar dynamical invariants for homeomorphisms, like entropy and periodic point data, become more complex and permit multiple definitions. We briefly survey some of these and other related invariants in the setting of algebraic Zd-actions, showing how, even in settings where the natural entropy as a Zd-action vanishes, a powerful theory of directional entropy and periodic points can be built. An underlying theme is uniformity in dynamical invariants as the direction changes, and the connection between this theory and problems in number theory; we explore this for several invariants. We also highlight Fried's notion of average entropy and its connection to uniformities in growth properties, and prove a new relationship between this entropy and periodic point growth in this setting.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series B following peer review. The definitive publisher-authenticated version "Miles, R and Ward, T (2015) Directional uniformities, periodic points, and entropy. Discrete and Continuous Dynamical Systems - Series B, 20 (10). pp. 3525-3545. ISSN 1531-3492" is available online at: http://dx.doi.org/10.3934/dcdsb.2015.20.3525. |
Keywords: | Directional dynamics, directional entropy, expansive subdynamics, algebraic dynamics |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 14 Oct 2016 09:27 |
Last Modified: | 17 Jan 2018 14:10 |
Published Version: | http://dx.doi.org/10.3934/dcdsb.2015.20.3525 |
Status: | Published |
Publisher: | American Institute of Mathematical Sciences (AIMS) |
Identification Number: | 10.3934/dcdsb.2015.20.3525 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:103800 |