One-dimensional semiconjugacy revisited

Let f be a piecewise monotone interval map with positive topological entropy h(f) = log(s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.