Homoclinic tangency of codimension 2 and historic wandering domains

TL;DR

This paper constructs a four-dimensional dynamical system with a high-codimension homoclinic tangency that produces a wandering domain where orbits display historic behavior, highlighting complex statistical phenomena in such systems.

Contribution

It provides the first example of a high-codimension homoclinic tangency leading to a positive measure wandering domain with non-convergent time averages.

Findings

Existence of a positive measure wandering domain with historic behavior.

Homoclinic tangencies of maximal codimension can generate complex statistical dynamics.

Orbits in the wandering domain do not have convergent time averages.

Abstract

We construct a four-dimensional diffeomorphism exhibiting a homoclinic tangency of the largest codimension, which admits a historic wandering domain of positive Lebesgue measure. Every orbit in this wandering domain exhibits historic behavior, in the sense that time averages do not converge. This example shows that homoclinic tangencies of the largest codimension can still give rise to positive Lebesgue measure sets with non-convergent statistical behavior.