Polynomial entropy on the $n$-fold symmetric product and its suspension

TL;DR

This paper investigates how polynomial entropy behaves under symmetric product and suspension operations for homeomorphisms with finite chain recurrent sets, establishing exact values and bounds.

Contribution

It proves that polynomial entropy scales linearly with the number of symmetric products and provides bounds for the suspension case under specific conditions.

Findings

Polynomial entropy of the induced map on the n-fold symmetric product equals n times the original.

Polynomial entropy of the suspension map equals n times the original polynomial entropy.

Provides a lower bound for polynomial entropy on the suspension when the homeomorphism has wandering points.

Abstract

We prove that the polynomial entropy of the induced map $F_n(f)$ on the $n$-fold symmetric product of a compact space $X$ and its suspension are both equal to $nh_{pol}(f)$, when $f:X\to X$ is a homeomorphism with a finite chain recurrent set $\mathcal{CR}(f)$. We also give a lower bound for the polynomial entropy on the suspension, for a homeomorphism $f$ with at least one wandering point, under certain assumptions.