A threshold for full packing dimension of Hölder images of sets and measures

Abstract

We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles ${\dim }_{P,\theta }^s\mu$, indexed by $\theta \in (0,1]$ and $s>0$, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at $\theta =1$. For a large family of random $\alpha$-Hölder maps $f_{\omega }:{\mathbb{R}}^n\to {\mathbb{R}}^m$, which includes index-$\alpha$ fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure $\mu$ on ${\mathbb{R}}^n$, $${\dim }_P{\mu }_{f_{\omega }}=m\text{almost surely}⟺\alpha m\le \underset{\theta \to 0}{\lim }{\dim }_{P,\theta }^n\mu .$$ We further study the profiles ${\dim }_P^s\mu$ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as $\theta \to 0$. Finally, we obtain the analogous characterization for analytic sets: $${\dim }_P{f}_{\omega }(E)=m\text{almost surely}⟺\alpha m\le \underset{\theta \to 0}{\lim }{\dim }_{P,\theta }^{n}E,$$ where ${\dim }_{P,\theta }^{n}E=\sup \{{\dim }_{P,\theta }^n\mu :\mu \in {\mathcal{M}}_c^{+}(E)\}$.