Non-statistical behavior via Statistical instability: Non-statistical Anosov-Katok diffeomorphisms
Amin Talebi
TL;DR
The paper addresses the emergence of non-statistical dynamics in dissipative Anosov-Katok diffeomorphisms of the annulus by formulating a general notion of statistical instability and linking it to non-statistical behavior. It develops an abstract framework with $\Delta_N$ and $\Delta$ to quantify sensitivity of empirical measures to perturbations and proves that non-statistical maps form a Baire generic subset inside the interior of statistically unstable maps. By combining this framework with the Anosov-Katok construction, it produces dissipative AK maps that are maximally oscillating and non-statistical, and shows that conservative approximants of AK maps (via Fayad–Katok) yield a precise ergodic-measure structure supporting this phenomenon. The work offers a strategy to identify non-statistical dynamics across other families and discusses connections to wandering domains and persistent behavior questions, highlighting the broader impact on understanding statistical properties in dynamical systems.
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.