Directional uniformities, periodic points, and entropy

Miles, Richard; Ward, Thomas

doi:10.3934/dcdsb.2015.20.3525

Directional uniformities, periodic points, and entropy

Miles, Richard; Ward, Thomas

Authors

Richard Miles

Thomas Ward

Abstract

Dynamical systems generated by d≥2 commuting homeomorphisms (topological Z d -actions) contain within them structures on many scales, and in particular contain many actions of Z k 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 Z d -actions, showing how, even in settings where the natural entropy as a Z d -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.

Citation

Miles, R., & Ward, T. (2015). Directional uniformities, periodic points, and entropy. Discrete and Continuous Dynamical Systems - Series B, 20(10), 3525-3545. https://doi.org/10.3934/dcdsb.2015.20.3525

Journal Article Type	Article
Acceptance Date	Mar 1, 2015
Publication Date	Dec 1, 2015
Deposit Date	Oct 2, 2015
Publicly Available Date	Sep 1, 2016
Journal	Discrete and Continuous Dynamical Systems - Series B
Print ISSN	1531-3492
Electronic ISSN	1553-524X
Publisher	American Institute of Mathematical Sciences (AIMS)
Peer Reviewed	Peer Reviewed
Volume	20
Issue	10
Pages	3525-3545
DOI	https://doi.org/10.3934/dcdsb.2015.20.3525
Keywords	Directional dynamics, Directional entropy, Expansive subdynamics, Algebraic dynamics.
Related Public URLs	http://arxiv.org/abs/1411.5295