Miles, Richard; Ward, Thomas B. Uniform periodic point growth in entropy rank one. (English) Zbl 1128.22004 Proc. Am. Math. Soc. 136, No. 1, 359-365 (2008). Let \(\alpha\) be an action of \({\mathbb Z}^d\) by continuous automorphisms of a compact metrizable abelian group \(X\) (such a system is called an algebraic \({\mathbb Z}^d\)-action). For a continuous map \(\beta : X\rightarrow X\) write \(h(\beta )\) for the topological entropy and \(F(\beta )=\{ x\in X | \beta x=x\}\) for the set of fixed points. The action \(\alpha\) is said to have entropy rank one if \(\forall n\in {\mathbb Z}^d\), \(h(\alpha^n )<\infty\). The main result of the paper: Let \(\alpha \) be a mixing algebraic \({\mathbb Z}^d\)-action with entropy rank one on a finite dimensional group \(X\). Then \(\exists C_{1} , C_2 \geq 0\) such that \(\limsup_{\rightarrow\infty} \frac{1}{| | x| | }\log | F(\alpha^n )| =C_1 <\infty \) and \(\liminf_{n\rightarrow \infty}\frac{1}{| | n| | }\log (F(\alpha^n )| =C_2\). If \(\dim(X)>0\) and the action is Noetherian, then \(C_2 >0\). It is shown that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions. Reviewer: Victor Sharapov (Volgograd) Cited in 3 Documents MSC: 37A15 General groups of measure-preserving transformations and dynamical systems 22D40 Ergodic theory on groups 37A35 Entropy and other invariants, isomorphism, classification in ergodic theory 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) Keywords:Algebraic \({\mathbb Z}^d\)-actions; entropy rank one; periodic point growth PDFBibTeX XMLCite \textit{R. Miles} and \textit{T. B. Ward}, Proc. Am. Math. Soc. 136, No. 1, 359--365 (2008; Zbl 1128.22004) Full Text: DOI arXiv References: [1] Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. · Zbl 0297.10013 [2] P. E. Blanksby and H. L. 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