Shift maps and statistical invariants for some dynamical systems

TL;DR

This paper introduces a new perspective on bifurcations and chaos in dynamical systems through shift functions, providing partial solutions to open problems and analyzing replicator maps related to evolutionary game dynamics.

Contribution

It presents a novel approach using shift maps to study invariants in dynamical systems, solving an open problem about maps with uniform periodic orbit averages and exploring replicator map dynamics.

Findings

Existence of hyperbolic chaos in studied maps

Continuous families of maps with all periodic orbits sharing the same mean value

Partial solution to an open problem on one-dimensional maps

Abstract

Given a dynamical system, we study the so-called space of shift functions thus introducing another vision on bifurcations and chaos. As an application of the obtained results, we give a partial solution to an open problem formulated in \cite{Misiurewicz1}: to describe all the one-dimensional maps with all the periodic orbits having the same mean value. Moreover, we show that there are continuous families of such mappings having infinitely many periodic points. For this purpose, we study the dynamics of the so-called replicator maps, depending on two parameters. Such studies are also motivated by the analysis of the dynamics of evolutionary games under selection. We prove the existence of hyperbolic chaos for the considered map and demonstrate that the average values are the same for all the periodic orbits.