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

TL;DR

This paper explores the relationship between P-configurations in dynamical systems and solutions to a specific functional equation, revealing conjugacy properties and the existence of continuous nonlinear solutions to a conditional Cauchy equation.

Contribution

It establishes conjugacy relations among P-configurations and proves the existence of continuous nonlinear solutions to a particular homogeneous functional equation.

Findings

Any two regular P-configurations are conjugate by a homeomorphism.

Regular P-configurations are not conjugate by a diffeomorphism.

Continuous nonlinear solutions exist for the specified functional 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 (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] .