Non-statistical behavior via Statistical instability: Non-statistical Anosov-Katok diffeomorphisms

TL;DR

This paper explores the concept of statistical instability as a key property behind non-statistical dynamics, introducing new non-statistical Anosov-Katok diffeomorphisms on the annulus and formalizing the notions of stability and instability.

Contribution

It formalizes statistical instability and constructs a new class of non-statistical Anosov-Katok diffeomorphisms demonstrating this property.

Findings

Existence of non-statistical maps linked to statistical instability.

Formalization of stability and instability notions.

Introduction of new non-statistical Anosov-Katok diffeomorphisms.

Abstract

\textit{Non-statistical dynamics} are those for which a set of points with positive measure (w.r.t. a reference probability measure which is in most examples the Lebesgue on a manifold) do not have a convergent sequence of empirical measures. In this paper, we show that behind the existence of non-statistical dynamics, there is some other dynamical property: \textit{statistical instability}. To this aim, we present a general formalization of the notions of statistical stability and instability and introduce sufficient conditions on a subset of dynamical systems to contain non-statistical maps in terms of statistical instability. We follow this idea and introduce a new class of non-statistical maps in the space of Anasov-Katok diffeomorphisms of the annulus.