Let \(X\) be a compact metrizable space and \(T:X\to X\) be a homeomorphism without fixed points. Then for each prime \(p\) the group \(\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}\) acts freely on the space \(P_p(X,T)\) of \(p\)-periodic points. The authors prove that the \(\mathbb{Z}_p\)-index \(\mathrm{ ind}_p(P_p(X,T))\) grows at most linearly in \(p\).
For the proof the authors introduce the spaces\[{\mathcal X}(N,\delta)=\left\{(x_n)_{n\in\mathbb{Z}}\in\big([0,1]^N\big)^{\mathbb{Z}}:|x_n-x_{n+1}|\ge\delta\right\},\qquad N\in\mathbb{N},\quad \delta>0,\]on which the shift map \(\sigma\) acts without fixed points. They show that \(\mathrm{ ind}_p\big(P_p({\mathcal X}(N,\delta))\big)\) grows at most linearly in \(p\), and they construct a continuous equivariant map \(X\to{\mathcal X}(N,\delta)\) for some \(N\in\mathbb{N}\), \(\delta>0\).
The spaces \({\mathcal X}(N,\delta)\) are also the main building blocks in the construction of a compact space \(X\) and a homeomorphism \(T\) on \(X\) without periodic points, and such that the dynamical system \((X,T)\) does not have the marker property (introduced in [Y. Gutman, Proc. Lond. Math. Soc. (3) 111, No. 4, 831–850 (2015;Zbl 1352.37017)]).
The paper concludes with some open problems concerning the marker property.

Reviewer: Thomas J. Bartsch (Gießen)

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