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Directional uniformities, periodic points, and entropy. (English) Zbl 1347.37035

Unlike dynamical systems generated by a single homeomorphism, a \(\mathbb{Z}^d\)-action which is generated by \(d\) commuting homeomorphisms, \(d\geq 2\), always exhibits much more complicated dynamics. Subdynamics of such an action, especially expansive subdynamics, were systematically investigated by M. Boyle and D. Lind [Trans. Am. Math. Soc. 349, No. 1, 55–102 (1997; Zbl 0863.54034)]. In particular, when considering one-dimensional subdynamics, the directional entropy is studied.
In this paper, the authors survey some of these and other related invariants in the setting of algebraic \(\mathbb{Z}^d\)-actions. They explore the uniformity in dynamical invariants as the direction changes, and the connection between this theory and problems in number theory. They 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.

MSC:

37B40 Topological entropy
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37P35 Arithmetic properties of periodic points
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)

Citations:

Zbl 0863.54034
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References:

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