A threshold for full packing dimension of H\"{o}lder images of sets and measures

Abstract

We study when the image of a measure under a random H\"{o}lder 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 $\alpha$-H\"{o}lder 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 $\mathbb{R}^n$, $$\dim_P \mu_{f_\omega}=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n \mu.$$ 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: $$\dim_P f_\omega(E)=m \quad\text{almost surely} \quad\Longleftrightarrow\quad \alpha m\leq \lim_{\theta\to 0}\dim_{P,\theta}^n E,$$ where $\dim_{P,\theta}^n E=\sup\{\dim_{P,\theta}^n \mu :\, \mu\in \mathcal{M}_c^+(E) \}$.