Berger's conjecture on local typicality of high emergence

Let $M$ be a compact manifold and let the emergence of a non-conservative map $f$ at scale $\epsilon>0$ be the least number of probability measures needed to approximate the empirical measures of Lebesgue almost every orbit at that scale. The map has high emergence when this quantity grows rapidly as $\epsilon\to 0$, in particular with maximal or otherwise super-polynomial order.

Berger's high-emergence conjecture. In many spaces of dynamical systems, high emergence is locally typical, in many senses.

The conjecture concerns the prevalence of statistically complex dynamics even when convergence of empirical measures may fail. It was proposed by Berger, and the source gives no resolution.