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A dichotomy in orbit growth for commuting automorphisms. (English) Zbl 1193.37006

The authors consider orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. The focus of the paper is on actions with an exponential rate of orbit growth \(g>0\). The topological entropy \(h\) is a global measure of orbit complexity, and dichotomy, studied in the paper, and concerning the relationship between \(g\) and \(h\). In the case \(g>h\) for a \({\mathbb Z}^2\) action, there is a preferred direction in which thin rectangular orbit shapes have an abundance of periodic orbits, and these dominate the count to such an extent that the orbit-counting asymptotics resemble the case of a single transformation. In the case \(g=h\) there are no preferred directions, and distinctly higher-dimensional asymptotics can arise. Many examples are given.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
22D40 Ergodic theory on groups
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