# Monotonicity of entropy for real quadratic rational maps

## TL;DR

This paper studies how the entropy of real quadratic rational maps varies across different regions of their moduli space, proving connectedness of entropy level sets in some regions and conjecturing monotonicity properties.

## Contribution

It proves connectedness of entropy level sets in certain regions and conjectures monotonicity behavior across the entire moduli space of real quadratic rational maps.

## Key findings

- Connected level sets of entropy in bimodal and part of unimodal regions.
- Conjecture that entropy is monotonic in the unimodal region.
- Conjecture that entropy monotonicity fails in the (+-+)-bimodal region.

## Abstract

The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$. Based on the numerical evidence, we conjecture that the monotonicity holds throughout the unimodal region, but we conjecture that it fails in the region of $(+-+)$-bimodal maps.