Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

TL;DR

This paper introduces a topological 'Stretching Along the Paths' method to find fixed and periodic points in expansive-contractive maps, revealing complex dynamics like chaos and entropy in Euclidean spaces, with applications in economics and biology.

Contribution

The paper presents a novel, accessible topological technique for detecting complex dynamics in continuous maps, avoiding advanced topological theories.

Findings

Proves semi-conjugacy to Bernoulli shift

Detects topological transitivity and sensitivity

Establishes positivity of topological entropy

Abstract

In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Our technique is also significant from a dynamical point of view, as it allows to detect complex dynamics. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover, our approach, although mathematically rigorous, avoids the use of sophisticated topological theories and it is relatively easy to apply to specific models arising in the applications. For example we have here employed the Stretching along the paths method to study discrete and continuous-time models arising from economics and biology.