Divergent coindex sequence for dynamical systems (Q6611243)

This paper advances the study of topological dynamics via the \(\mathbb{Z}_p\)-index theory initiated in [\textit{M. Tsukamoto} et al., Isr. J. Math. 251, No. 2, 737--764 (2022; Zbl 1518.37023)].\N\NGiven a topological dynamical system \((X, T)\), i.e., a compact metrizable space \(X\) and a homeomorphism \(T : X \to X\), the set of \(n\)-periodic points of \((X, T)\), for \(n \geq 1\), is defined as\N\[\NP_n(X) := \{x \in X : T^n x = x\}.\N\]\NSuch a system is called fixed-point free if \(P_1(X) = \emptyset\), meaning that it has no fixed points.\N\NFor each prime number \(p\), if \((X, T)\) is fixed-point free, then the pair \((P_p(X), T)\) is a free \(\mathbb{Z}_p\)-space. In [loc. cit.], the authors analyzed this setting and investigated the \(\mathbb{Z}_p\)-index of \(P_p(X)\). Roughly, the index and coindex of a freely acts on a topological space measure the size of the given action.\N\NThe contribution of the current paper is an affirmative solution to a problem posed in [loc. cit.], namely whether there exists a fixed-point-free dynamical system \((X,T)\) such that the sequence \((\operatorname{ind}_p P_p(X))\), indexed by prime numbers \(p\), is unbounded. The proof leverages the marker property, and from this, the author exhibit some other fixed-point free dynamical systems with divergent coindex sequences.