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

Abstract

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 entropy dimension of a self-conformal measure exist, without assuming any separation condition. 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. We also generalize the result on the existence of the $L^q$-dimension to self-conformal measures on complete Riemannian manifolds with the doubling property. In particular, these results hold for complete Riemannian manifolds with nonnegative Ricci curvature. Moreover, by assuming that the measure is doubling, we extend the result on the existence of the entropy dimension to self-conformal measures on complete Riemannian manifolds.