Determiniation of the homotopy class of maps on compact orientable surfaces of positive genus g with infinitely many periodic points (part I)

TL;DR

This paper provides an algebraic criterion for identifying homotopy classes of maps on compact orientable surfaces with positive genus that have infinitely many periodic points, using homology and fixed point theory instead of Nielsen numbers.

Contribution

It introduces a new algebraic approach based on the characteristic polynomial of homomorphisms for analyzing periodic points, differing from traditional Nielsen number methods.

Findings

Established a necessary and sufficient algebraic criterion for infinite periodic points

Applied fixed point theory and intersection theory in a novel way for surface dynamics

Reproduced known results for the torus case using the new approach

Abstract

We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms of the characteristic polynomial of the matrix representing the correspondig homomorphism of the first homology group. Our methods differ from that of the literature, which typically uses Nielsen number theory, because Nielsen numbers of iterates are difficult to compute except in special cases where they can be easily defined in terms of the easily computable Lefschetz number. Instead we use some fixed point theory from the point of view of real algebraic geometry, and some intersection theory in the context of De Rham theory on surfaces and combine it with some observations of dynamical system theory which started from the work of Sarkovskii. However, in the special case of the homotopical dynamics of a two dimensional torus, where the Nielsen number is determined by the modulus of the Lefschetz number, we recover a result in [1].