Second Order Ergodic Theorem for Self-Similar Tiling Systems
  We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule. 
Konstantin Medynets and Boris Solomyak
math.DS