Monotonicity of entropy for real multimodal maps

TL;DR

This paper proves Milnor's Monotonicity Conjecture, establishing that in families of real multimodal polynomial interval maps, the set of parameters with constant topological entropy is always connected.

Contribution

It generalizes previous results by proving the conjecture for all real multimodal polynomial maps, not just quadratic or cubic cases.

Findings

Connectedness of entropy level sets for all real multimodal polynomial maps

Extension of Milnor & Thurston's quadratic case proof to general multimodal maps

Confirmation of the Monotonicity Conjecture in the general setting

Abstract

In \cite{Mil}, Milnor posed the {\em Monotonicity Conjecture} that the set of parameters within a family of real multimodal polynomial interval maps, for which the topological entropy is constant, is connected. This conjecture was proved for quadratic by Milnor & Thurston \cite{MT} and for cubic maps by Milnor & Tresser, see \cite{MTr} and also \cite{DGMT}. In this paper we will prove the general case.