Second-order ergodic theorem for self-similar tiling systems
Second-order ergodic theorem for self-similar tiling systems
Abstract 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.