Dyadic covariance condition for the entire balanced map class

Establish the dyadic covariance estimate for every balanced doubly intermittent map g in the class \(\mathfrak F_*\), namely, for a countable uniformly dense family of observables \(\mathcal D\subset C([-1,1])\), prove that each \(\phi\in\mathcal D\) satisfies a power-saving bound of the form \(\sum_{0\le i,j<n}|\operatorname{Cov}_m(\phi\circ g^{2^i},\phi\circ g^{2^j})|\le C_\phi n^{2-\delta_\phi}\) for some \(C_\phi<\infty\) and \(\delta_\phi>0\), thereby establishing almost-sure convergence of the dyadic empirical measures for every map in \(\mathfrak F_*\).

Background

The paper develops an abstract variance criterion showing that almost-sure convergence of sparse empirical measures follows from a power-saving variance bound, and it gives a sufficient double-covariance estimate for obtaining that variance bound. For dyadic sampling $a_k=2^k$, the strong natural-measure property supplies the required one-time convergence, but not the necessary two-time covariance control.

The authors verify the required estimate for the symmetric quadratic map $g_1$ on a countable uniformly dense family of endpoint-flat observables, obtaining an $O_\phi(n\log(n+2))$ bound and consequently almost-sure dyadic convergence for $g_1$. The unresolved problem is to extend this verification from $g_1$ to every map in the balanced class $\mathfrak F_*$.