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}.

34 pages; 4 figures; this version focuses on abundance of wild historic points and leaves the ergodic decomposition and generalized physical measures part for another publication; updated with referees comments and suggestions. To appear in Bull Braz Math Soc

References in corpus (5)

• Takens' last problem and existence of non-trivial wandering domains
• On Takens' Last Problem: tangencies and time averages near heteroclinic networks
• Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets
• Geometric Lorenz flows with historic behavior
• Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Cited by in corpus (3)

• Highly irregular orbits for subshifts of finite type: large intersections and emergence
• Lyapunov exponents for random maps
• Irregular sets for piecewise monotonic maps