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

TL;DR

This paper extends the existence and equivalence of $L^q$-dimension and entropy dimension for self-conformal measures from Euclidean spaces to complete Riemannian manifolds with the doubling property, including those with nonnegative Ricci curvature.

Contribution

It introduces heavy maximal packings and partitions to prove the equivalence of $L^q$-dimension and generalized dimension on doubling metric spaces, and extends these results to Riemannian manifolds.

Findings

$L^q$-dimension exists for self-conformal measures on Riemannian manifolds.

$L^q$-dimension is equivalent to the generalized dimension on doubling metric spaces.

Entropy dimension exists for self-conformal measures on Riemannian manifolds with doubling measures.

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.