On the irregular points for systems with the shadowing property

Abstract

We prove that when f is a continuous self-map acting on a compact metric space (X, d) which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy.Using this fact we prove that in the class of C 0 -generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergent Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak * topology. Moreover, such points carry zero entropy. In contrast, irregular points are nonobservable but carry infinite entropy.

Properties of invariant measures in dynamical systems with the shadowing property

“…The following result is a simplified, one-dimensional version of [26, Proposition 2.1] (cf. [12,11]). It will be very useful in further calculations in Section 4.…”

Distributional chaos in multifractal analysis, recurrence and transitivity

“…Briefly, multifractal analysis studies the dynamical complexity of the level sets of the invariant local quantities obtained from a dynamical system. There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc., for example, see [52,9,51,16,67,23,5,66,28] (for topological entropy or Hausdorff dimension), [68,69] (for topological pressure), [64,38] (for Lebesgue positive measure) and references therein. However, it is unknown from the viewpoint of chaos.…”

“…In other words, the orbits of x and y are arbitrarily close with upper density one, but for some distance, with lower density zero. ϕ-regular set and the irregular set, the union of I ϕ (f ) over all continuous functions of ϕ (denoted by IR(f )), arise in the context of multifractal analysis and have been studied a lot, for example, see [52,9,51,16,69,23]. The irregular points are also called points with historic behavior, see [57,64].…”

“…However, the irregular set may have strong dynamical complexity in sense of Hausdorff dimension, Lebesgue positive measure, topological entropy and topological pressure etc.. Pesin and Pitskel [52] are the first to notice the phenomenon of the irregular set carrying full topological entropy in the case of the full shift on two symbols. There are lots of advanced results to show that the irregular points can carry full entropy in symbolic systems, hyperbolic systems, non-uniformly expanding or hyperbolic systems, and systems with specification-like or shadowing-like properties, for example, see [9,51,16,69,23,46,71]. For topological pressure case see [69] and for Lebesgue positive measure see [64,38].…”

The conditional variational principle for maps with the pseudo-orbit tracing property

“…The property led to fruitful results in the study of ergodic theory and qualitative theory of dynamical systems (see [1] and [5]). Recently, Dong, Oprocha and Xueting Tian [6] showed that the set of points with divergent Birkhoff averages is either empty or carries full topological entropy. Let…”