Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously

Abstract

We prove a universal projection theorem, giving conditions on a parametrized family of maps $\Pi_\lambda : X \to \mathbb{R}^d$ and a collection M of measures on X under which for almost every $\lambda$ equality $\dim_H \Pi_\lambda \mu = \min\{d, \dim_H \mu\}$ holds for all measures $\mu \in M$ simultaneously (i.e. on a full measure set of $\lambda$'s independent of $\mu$). We require family $\Pi_\lambda$ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every $\lambda$, projection $\Pi_\lambda$ is nearly bi-Lipschitz (i.e. with pointwise $\alpha$-H\"older inverse for every $\alpha \in (0,1)$) at $\mu$-a.e. x, for all $\mu \in M$ simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS. As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part). Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation).