Abundance of wild historic behavior

TL;DR

This paper investigates wild historic points in dynamical systems using Caratheodory measures, establishing their generic abundance and linking their properties to system features like heteroclinic connections.

Contribution

It introduces a measure-theoretic approach to study historic points, constructing topologically generic sets of wild historic points for various dynamical models.

Findings

Existence of invariant measures associated to historic points.

Topologically generic sets of wild historic points are constructed.

Connections between historic points and heteroclinic structures are established.

Abstract

Using Caratheodory measures, we associate to each positive orbit ${\mathcal O}_{f}^{+}(x)$ of a measurable map $f$, a Borel measure $\eta_{x}$. We show that $\eta_{x}$ is $f$-invariant whenever $f$ is continuous or $\eta_{x}$ is a probability. These measures are used to study the \emph{historic} points of the system, that is, \emph{points with no Birkhoff averages}, and we construct topologically generic subset of \emph{wild historic points} for wide classes of dynamical models. We use properties of the measure $\eta_x$ to deduce some features of the dynamical system involved, like the \emph{existence of heteroclinic connections from the existence of open sets of historic points}.