Miles, Richard; Ward, Thomas B. A dichotomy in orbit growth for commuting automorphisms. (English) Zbl 1193.37006 J. Lond. Math. Soc., II. Ser. 81, No. 3, 715-726 (2010). 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. Reviewer: Victor Sharapov (Volgograd) Cited in 1 ReviewCited in 3 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 22D40 Ergodic theory on groups Keywords:group actions; topological entropy; exponential rate of orbit growth PDFBibTeX XMLCite \textit{R. Miles} and \textit{T. B. Ward}, J. Lond. Math. Soc., II. Ser. 81, No. 3, 715--726 (2010; Zbl 1193.37006) Full Text: DOI arXiv Link