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

TL;DR

This paper establishes a universal projection theorem under transversality and relative dimension separability conditions, enabling simultaneous dimension results for measures and projections in multifractal and self-conformal contexts.

Contribution

It introduces the relative dimension separability condition and proves a universal projection theorem applicable to various families of maps and measures, advancing multifractal analysis and dimension theory.

Findings

Dimension equality holds for almost every parameter and all measures in the collection.

Projection maps are nearly bi-Lipschitz under certain dimension conditions.

New formulas for Hausdorff dimension of level sets of local dimensions.

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).