Dynamical blanket times

Abstract

We introduce dynamical blanket times, which quantify how quickly the empirical distribution along a typical orbit approximates the invariant measure. These can be viewed as a measure-theoretic analogue of the previously introduced \emph{dynamical cover time}, which measures how quickly an orbit becomes dense in the space. Motivated by analogous comparability questions for random walks on graphs, we investigate how much longer it takes for a dynamical system to ``blanket'' than to ``cover''. For finite-branch, uniformly expanding interval maps, we obtain upper bounds on the expected blanket time in terms of the spatial scale and the precision of approximation. In the special case where the invariant measure is absolutely continuous with respect to Lebesgue, this yields comparability between the expected blanket and cover times, uniformly across all sufficiently small scales. Our approach combines two main ingredients. First, we establish large deviation estimates for hitting times which are uniform over both the target location and the spatial scale. Second, using methods from multifractal analysis, we construct a finite discretisation of the invariant measure which reduces the problem to a suitably controlled discrete model.