Sparse empirical measures for non-statistical interval maps with two neutral fixed points

Abstract

We consider balanced doubly intermittent full-branch interval maps with two neutral fixed points. Although the ordinary empirical measures fail to converge for Lebesgue-almost every initial point, we show that this non-statistical behavior persists under a broad class of deterministic observation schemes: if the sampling sequence is block-dominating, then both endpoint Dirac masses are accumulation points of the sparse empirical measures for Lebesgue-almost every orbit. This class includes sampling sequences of positive lower density, arithmetic and polynomial sequences, and finite-index regularly varying sequences. We also prove that every map in the balanced class has the strong natural-measure property, including at the boundary exponent $β=1$: the pushforwards of every absolutely continuous initial probability converge to a distinguished endpoint mixture $ν_*$. Abstract consequences of this one-time limit include annealed convergence along every deterministic sampling sequence, the existence of arbitrarily sparse sequences with almost-sure convergence, and an almost-sure convergence criterion based on a power-saving variance bound. For the symmetric quadratic map $g_1\in\mathfrak F_*$, we verify the dyadic covariance condition on a countable uniformly dense endpoint-flat family: the double covariance sum is $O_ϕ(n\log(n+2))$. Consequently its dyadic empirical measures converge for Lebesgue-almost every initial point to $\tfrac12(δ_{-1}+δ_1)$.