On the non-monotonicity of entropy for a class of real quadratic rational maps

TL;DR

This paper demonstrates that the entropy function for real quadratic rational maps is non-monotonic, contrasting polynomial cases, by showing disconnected level sets and confirming a conjecture for bimodal maps.

Contribution

It proves the non-monotonicity of entropy in a specific class of real quadratic rational maps, confirming a previously conjectured behavior.

Findings

Entropy function is non-monotonic for these maps.

Level sets of entropy are disconnected.

Confirms conjecture on bimodal shape maps.

Abstract

We prove that the entropy function on the moduli space of real quadratic rational maps is not monotonic by exhibiting a continuum of disconnected level sets. This entropy behavior is in stark contrast with the case of polynomial maps, and establishes a conjecture on the failure of monotonicity for bimodal real quadratic rational maps of shape $(+-+)$ which was posed in arXiv:1901.03458 based on experimental evidence.