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arXiv · 1609.05356

Abundance of wild historic behavior

Vitor Araujo·Vilton Pinheiro
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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}.

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Vitor Araujo, Vilton Pinheiro. 2016-09-17. Abundance of wild historic behavior. https://doi.org/10.1007/s00574-019-00191-8

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