Polynomial entropy on regular curves
Maša Đorić, Jelena Katić
TL;DR
The paper investigates polynomial entropy $h_{\mathrm{pol}}$ for homeomorphisms on regular curves, showing it is bounded by a quantity that grows at most logarithmically with the preimage-component count and that equality with 1 occurs precisely when wandering behavior is present. It proves a general bound $h_{\mathrm{pol}}(f)\le 1+\limsup_{n}\frac{\log\varphi(f,n)}{\log n}$ on regular curves, implies $h_{\mathrm{pol}}(f)\le 1$ for all regular-curve homeomorphisms, and identifies a wandering-point criterion giving $h_{\mathrm{pol}}(f)=1$. For local dendrites, a rigidity result ensues: $h_{\mathrm{pol}}(f)\in\{0,1\}$ with $1$ exactly when wandering occurs. The authors also analyze induced dynamics on hyperspaces, establishing that $h_{\mathrm{pol}}(F_n(f))=n$ under wandering and that $h_{\mathrm{pol}}(2^f)=0$ when wandering is absent, thereby linking polynomial entropy to equicontinuity in these 1D continua.
Abstract
We show that the polynomial entropy of homeomorphisms on regular curves is bounded above by one. Moreover, the polynomial entropy equals one under the fairly mild condition that the homeomorphism possesses a wandering point. We obtain a rigidity result for homeomorphisms on local dendrites (and therefore on graphs and dendrites): their polynomial entropy is either zero or one.