  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.
