Non-statistical rational maps

Abstract

We show that in the family of degree $$d\geqslant 2$$ 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 equal to the largest possible one, that is the set of all f-invariant measures. We also introduce an abstract setting to study non-statistical dynamics and sufficient conditions for their existence in a general family of maps. These sufficient conditions are related to the notions of “statistical instability” and “statistical bifurcation” for which we give a general formalization in the first section. The proof of our result in the family of rational maps 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. Using this property and doing careful perturbations, we show that the closure of strictly postcritically finite maps satisfies the sufficient conditions introduced in the abstract setting.