Non-statistical rational maps

TL;DR

This paper demonstrates that within the family of degree d rational maps, a generic subset exhibits maximal non-statistical behavior, with orbits' empirical measures accumulating on all invariant measures, revealing complex dynamical properties.

Contribution

It introduces the concept of statistical bifurcation and shows that the closure of strictly postcritically finite maps contains a generic set with maximal non-statistical behavior.

Findings

Generic maps have empirical measures accumulating on all invariant measures.

The proof uses transversality to control critical orbit behavior.

Introduces the concept of statistical bifurcation.

Abstract

We show that in the family of degree $d\geq 2$ rational maps of the Riemann sphere, the closure of strictly postcritically finite maps contains a (relatively) Baire generic subset of maps displaying maximal non-statistical behavior: for a map $f$ in this generic subset, the set of accumulation points of the sequence of empirical measures of almost every point in the phase space is the largest possible one that is, the set of all $f$-invariant measures. The proofs is based on a transversality argument which allows us to control the behavior of the orbits of critical points for maps close to strictly postcritically finite rational maps and also a new concept developed in the author's PhD thesis, that we call statistical bifurcation.