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 $\dim_{P,θ}^{s}μ$, indexed by $θ\in(0,1]$ and $s>0$, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at $θ=1$. For a large family of random $α$-Hölder maps $f_ω:\mathbb{R}^n\to\mathbb{R}^m$, which includes index-$α$ fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure $μ$ on $\mathbb{R}^n$, $$\dim_P μ_{f_ω}=m \quad\text{almost surely} \quad\Longleftrightarrow\quad αm\leq \lim_{θ\to 0}\dim_{P,θ}^n μ.$$ We further study the profiles $\dim_P^sμ$ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as $θ\to 0$. Finally, we obtain the analogous characterization for analytic sets: $$\dim_P f_ω(E)=m \quad\text{almost surely} \quad\Longleftrightarrow\quad αm\leq \lim_{θ\to 0}\dim_{P,θ}^n E,$$ where $\dim_{P,θ}^n E=\sup\{\dim_{P,θ}^n μ:\, μ\in \mathcal{M}_c^+(E) \}$.