Persistent Non-Statistical Dynamics in One-Dimensional Maps

Abstract

We study a class $\hat{\mathfrak{F}}$ of one-dimensional full branch maps introduced in [Doubly Intermittent Full Branch Maps with Critical Points and Singularities; D. Coates, S. Luzzatto, M. Mubarak, 2022], admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that $\hat{\mathfrak{F}}$ can be partitioned into 3 pairwise disjoint subfamilies $$\hat{\mathfrak{F}}=\mathfrak{F}\cup {\mathfrak{F}}_{\pm }\cup {\mathfrak{F}}_{\ast }$$ such that all $g\in \mathfrak{F}$ have a unique physical measure equivalent to Lebesgue, all $g\in {\mathfrak{F}}_{\pm }$ have a physical measure which is a Dirac-$\delta$ measure on one of the (repelling) fixed points, and all $g\in {\mathfrak{F}}_{\ast }$ are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.

29 pages, 1 figure