Monotonicity of entropy for unimodal real quadratic rational maps

TL;DR

This paper proves that the topological entropy increases monotonically for a class of unimodal interval maps derived from real quadratic rational maps, confirming a previously conjectured behavior based on experimental data.

Contribution

It establishes the monotonicity of entropy for these maps by analyzing the moduli space and ruling out specific post-critical curves, confirming a conjecture from prior work.

Findings

Topological entropy is monotonic for the specified maps.

Certain post-critical curves do not exist in the moduli space.

The result confirms a conjecture based on experimental evidence.

Abstract

We show that the topological entropy is monotonic for unimodal interval maps which are obtained from the restriction of quadratic rational maps with real coefficients. This is done by ruling out the existence of certain post-critical curves in the moduli space of aforementioned maps, and confirms a conjecture made in [Fil19] based on experimental evidence.