On the structure of isentropes of real polynomials

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.