Almost-sure behaviour of empirical measures for intermittent maps

Interval maps with several equally sticky and sufficiently sticky neutral fixed points can present non-statistical behaviour where the sequence of empirical measures does not converge for Lebesgue almost every initial condition. If the order of the tangency of the map to the identity at the fixed points is small enough there is a unique physical measure (equivalent to Lebesgue) and the map is statistical. If instead the order of this tangency is sufficiently large, then the empirical measures almost surely accumulate to the full simplex of invariant measures supported at the fixed points and the map is non-statistical. I will present recent ongoing work with Tanja I. Schindler (University of Exeter) where we describe the almost sure behaviour of the empirical measures at the boundary case between these two phenomena.