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

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 $\mathrm{dim}_H \Pi_\lambda \mu = \min\{d, \mathrm{dim}_H \mu\}$ holds for all measures $\mu \in M$ simultaneously (i.e. on a full-measure set of $\lambda$'s independent of $\mu$). We require $\Pi_\lambda$ to satisfy a transversality condition and $M$ to satisfy a new condition called relative dimension separability. 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 at $\mu$-a.e. $x$, for all measures $\mu \in M$ simultaneously. Our setting can include families of orthogonal projections, natural projections for conformal, non-autonomous or random iterated functions systems. As an application, we prove that for a parametrized family of contracting conformal IFS with natural projections $\Pi_\lambda$ satisfying the transversality condition, for almost every parameter $\lambda$ one has $\mathrm{dim}_H \Pi_\lambda \mu = \min\{ d, \frac{h(\mu)}{\chi(\lambda, \mu)} \}$ for all ergodic shift-invariant measures simultaneously. We also prove that for self-similar systems on the line with similarity dimension smaller than one, for Lebesgue almost every choice of translations the multifractal formalism holds simultaneously on the full spectrum interval $\left[\min \frac{\log p_i}{\log|\lambda_i|},\max \frac{\log p_i}{\log|\lambda_i|}\right]$ for every self-similar measure. We prove that the dimension part of the Marstrand-Mattila projection theorem holds simultaneously for the collection of all ergodic measures on a self-conformal set with the strong separation condition and without any separation for the collection of all Gibbs measures.