Self-semiconjugation of piecewise linear unimodal maps

Abstract

We devote this work to the functional equation ψ • g = g • ψ, where ψ is an unknown function and g is piecewise linear unimodal map, which is topologically conjugated to the tent map. We will call such ψ self-semiconjugations of g. Our the main results are the following:1. Suppose that there is a self-semiconjugation of g, whose tangent at 0 is not a power of 2, and suppose that all the kinks of g are in the complete pre-image of 0. Then all the self-semiconjugations of g are piecewise linear.2. Suppose that all self-semiconjugations of g are piecewise linear. Then the conjugacy of g and the tent map is piecewise linear 1 2 .