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We consider completely invariant subsets A of weakly expanding piece-wise monotonic transformations T on 0;1]. It is shown that the upper box dimension of A is bounded by the minimum t A of all parameters t for which a t-conformal measure with support A exists. In particular, this implies equality of box dimension and Hausdorr dimension of A.
We extend the notions of Hausdorff and packing dimension introducing weights in their definition. These dimensions are computed for ergodic invariant probability measures of two-dimensional Lorenz transformations, which are transformations of the type occuring as first return maps to a certain cross section for the Lorenz differential equation. We give a… (More)