Abundance of Wild Historic Behavior

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}.