# On the structure of isentropes of real polynomials

## TL;DR

This paper modifies the Milnor--Thurston map to analyze the structure of isentropes of real polynomials, providing insights into entropy monotonicity and approximation by hyperbolic maps.

## Contribution

It introduces a modified Milnor--Thurston map to study isentropes, offering a new proof of entropy monotonicity and characterizing maps that cannot be approximated by hyperbolic maps.

## Key findings

- Proves monotonicity of topological entropy for real polynomials.
- Identifies maps with specific combinatorics that cannot be approximated by hyperbolic maps.
- Provides a framework for understanding the structure of isentropes.

## Abstract

In this paper we will modify the Milnor--Thurston map, which maps a one dimensional mapping to a piece-wise linear of the same entropy, and study its properties. This will allow us to give a simple proof of monotonicity of topological entropy for real polynomials and better understand when a one dimensional map can and cannot be approximated by hyperbolic maps of the same entropy. In particular, we will find maps of particular combinatorics which cannot be approximated by hyperbolic maps of the same entropy.