Abundance of Wild Historic Behavior

Abstract

Using Caratheodory measures, we associate to each positive orbit O + f (x) of a measurable map f , a Borel measure η x . We show that η x is f -invariant whenever f is continuous or η x 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 η x to deduce some features of the dynamical system involved, like the existence of heteroclinic connections from the existence of open sets of historic points.

Emergence for diffeomorphisms with nonzero Lyapunov exponents

“…We complete the proof of the second statement of the lemma by observing that (2). In addition, by Theorem 3.2, we conclude…”

“…This result was later extended to topologically mixing subshifts of finite type in [5], to graph directed Markov systems in [16], to continuous maps with specification property in [13] (see also [38]), and to continuous maps with almost specification property in [39]. Other works related to the study of the set of irregular points include [18,34,36,1,11,12,40,10,22,2,4,41].…”

“…2 , s) , and for every given x ∈ Y and m ∈ N the corresponding subcollection I x,m := {(x, x 2 , . .…”

Sensitivity and historic behavior for continuous maps on Baire metric spaces

“…In the last decades, though, there has been an intense study concerning the set of points for which Cesàro averages do not converge. Contrary to the previous measure-theoretical description, the set of the irregular points may be Baire generic and, moreover, have full topological pressure, full metric mean dimension or full Hausdorff dimension (see [2,5,3,4,22,24,31]). In [8], the first and the fourth named authors obtained a simple and unifying criterion, using first integrals, to guarantee that I(T, ϕ) is Baire generic whenever T : X → X is a continuous dynamics acting on a compact metric space X.…”

Irregular sets for piecewise monotonic maps

“…Pesin-Pitskel's thermodynamic dichotomy for irregular sets was extended to topologically mixing subshifts of finite type in [3] (together with the detailed study of the set of points at which Lyapunov exponent or local entropy fail to exist), to continuous maps with specification property in [9] (see also [28]), and to continuous maps with almost specification property in [29]. See also [1,2,5,6,20,25,27] and references therein for the study of irregular sets from other viewpoints.…”