Dynamics of random selfmaps of surfaces with boundary

TL;DR

This paper investigates the behavior of selfmaps on surfaces with boundary, showing that most such maps have exponentially many periodic points of every period, with growth rates linked to topological entropy.

Contribution

It applies Nielsen theory and Wagner's algorithm to establish generic exponential growth of periodic points for selfmaps of surfaces with boundary.

Findings

Most selfmaps have periodic points of all periods.

Number of periodic points grows exponentially with period.

Results hold for any space homotopy equivalent to a surface with boundary.

Abstract

We use Wagner's algorithm to estimate the number of periodic points of certain selfmaps on compact surfaces with boundary. When counting according to homotopy classes, we can use the asymptotic density to measure the size of sets of selfmaps. In this sense, we show that "almost all" such selfmaps have periodic points of every period, and that in fact the number of periodic points of period n grows exponentially in n. We further discuss this exponential growth rate and the topological and fundamental-group entropies of these maps. Since our approach is via the Nielsen number, which is homotopy and homotopy-type invariant, our results hold for selfmaps of any space which has the homotopy type of a compact surface with boundary.