Abundance of Wild Historic Behavior

Using Caratheodory measures, we associate to each positive orbit [inline-graphic not available: see fulltext] of a measurable map f, a Borel measure ηx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{x}$$\end{document}. We show that ηx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{x}$$\end{document} is f-invariant whenever f is continuous or ηx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{x}$$\end{document} is a probability. These measures are used to study the historic points of the system, that is, points with no Birkhoff averages, and we construct topologically generic subset of wild historic points for wide classes of dynamical models. We use properties of the measure ηx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _x$$\end{document} to deduce some features of the dynamical system involved, like the existence of heteroclinic connections from the existence of open sets of historic points.