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

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 (resolving an open problem posed by Paneah): there exist continuous nonlinear solutions to the functional equation: f(t) = f((t+1)/2) + f((t-1)/2), t \in [-1,1] .