Quasi-Lipschitz mapping, correlation and local dimensions ☆

In this article, we discuss two important and related concepts in the studies of geometric dimension theory, e.g. the correlation dimension and the local dimension of measures. Our results can be summarized as the following two aspects: on one hand, we show that the correlation dimension of measures is invariant under the quasi-Lipschitz mapping, and also give a sufficient condition for the coincidence of the correlation dimension and the Hausdorff dimension of measures. On the other hand, we examine the local dimensions in the limit sets of Moran construction in abstract metric space, with reasonably weaker separation condition. These discussions generalized several known results by Mattila, Moran and Rey in [14] and Li, Lou and Wu in [10,11].