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Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the$L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropydimension of a self-conformal measure exist, without assuming any separationcondition. By introducing the notions of heavy maximal packings and partitions,we prove that on a doubling metric space the $L^q$-dimension,$q\in(0,\infty)\setminus\{1\}$, is equivalent to the generalized dimension. Wealso generalize the result on the existence of the $L^q$-dimension toself-conformal measures on complete Riemannian manifolds with the doublingproperty. In particular, these results hold for complete Riemannian manifoldswith nonnegative Ricci curvature. Moreover, by assuming that the measure isdoubling, we extend the result on the existence of the entropy dimension toself-conformal measures on complete Riemannian manifolds.

Existence of $L^q$-dimension and entropy dimension of self-conformal measures on Riemannian manifolds