A criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology

TL;DR

This paper introduces a new criterion for detecting periodic points in certain topological spaces by analyzing eigenvalues of induced cohomology maps, with examples and a counterexample to previous claims.

Contribution

It provides a novel eigenvalue-based criterion for periodic points in spaces with specific cohomology structures, expanding the tools for topological dynamics analysis.

Findings

Established a criterion linking eigenvalues to periodic points

Presented natural and non-trivial examples of spaces meeting the criterion

Countered a previous unproven claim in the literature

Abstract

We present a criterion for the existence of periodic points based on the eigenvalues of maps induced in cohomology for spaces with rational cohomology isomorphic to a tensor product of a graded exterior algebra with generators in odd dimensions and a graded algebra with all elements of even degree. We give a number of natural examples of such spaces and provide some non-trivial ones. We also give a counterexample to a claim in \cite{duan} given there without proof.