Asymptotic (statistical) periodicity in two-dimensional maps

TL;DR

This paper introduces a new sufficient condition for asymptotic periodicity of the Frobenius-Perron operator in two-dimensional maps, extending applicability to systems with eigenvalues less than one, and demonstrates this with a novel dynamical system.

Contribution

The paper presents a new theorem that broadens the class of two-dimensional maps for which asymptotic periodicity can be established, including systems with eigenvalues smaller than one.

Findings

Established a new sufficient condition for asymptotic periodicity.

Applied the theorem to a newly introduced two-dimensional system.

Demonstrated different periodic behaviors depending on parameters.

Abstract

In this paper we give a new sufficient condition for asymptotic periodicity of Frobenius-Perron operator corresponding to two--dimensional maps. The result of the asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical systems was already known. Our new theorem enables to apply for the system having an eigenvalue smaller than one. The key idea for the proof is a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system exhibiting the asymptotic periodicity with different periods depending on parameter values, and discuss to apply our theorem to the model.