Non-statistical behavior via statistical instability: non-statistical Anosov–Katok diffeomorphisms

Non-statistical (or historical) dynamics refer to systems where a set of points with positive measure, relative to a reference probability measure (typically the Lebesgue measure on a manifold), fails to exhibit a convergent sequence of empirical measures. There are various strategies to identify maps with non-statistical behavior. In the first part of this paper, we use a strategy related to statistical instability to demonstrate that a Baire generic map in the space of Anosov–Katok diffeomorphisms of the annulus exhibits non-statistical behavior and even maximal oscillation. The notion of statistical instability was first formalized in its general format by Talebi [Non-statistical rational maps. Math. Z. 302 (1) (2022), 589–608], where its close connection to the existence of non-statistical dynamics was established. In the second part, we extend this framework by introducing analogous formalization and corresponding results in the case of pointwise convergence of empirical measures. Additionally, we propose a general strategy to identify non-statistical dynamics within other families of dynamical systems.