Conjugacy of P-configurations and nonlinear solutions to a certain conditional Cauchy equation

Abstract

We study the connection between conjugations of a special kind of dynamical systems, called P-configurations, and solutions to homogeneous Cauchy type functional equations. We find that any two regular P-configurations are conjugate by a homeomorphism, but cannot be conjugate by a diffeomorphism. This leads us to the following conclusion (answering an open question posed by Paneah): there exist continuous nonlinear solutions to the functional equation: $$ f(t) = f\left(\frac{t+1}{2}\right) + f\left(\frac{t-1}{2}\right) \,\, , \,\, t \in [-1,1] . $$