Directional uniformities, periodic points, and entropy

Richard  Miles; Thomas  Ward

doi:10.3934/dcdsb.2015.20.3525

Article Contents

2015, Volume 20, Issue 10: 3525-3545. Doi: 10.3934/dcdsb.2015.20.3525

This issue Previous Article Entropy determination based on the ordinal structure of a dynamical system Next Article Topological mixing, knot points and bounds of topological entropy

Directional uniformities, periodic points, and entropy

Richard Miles^1, and
Thomas Ward^2,

1.
Uppsala Universitet, Lägerhyddsvägen 1, Hus 1, 5 och 7, 75106 Uppsala
2.
Durham University, Durham DH1 3LE

Received: October 31, 2014

Revised: February 28, 2015

Published: November 2015

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Abstract

Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. 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 $\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy as a $\mathbb{Z}^d$-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.

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Mathematics Subject Classification: Primary: 37B40, 37C25; Secondary: 37P35, 37C40.

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