Hausdorff dimension of shrinking targets on Przytycki-Urbański fractals
Thomas Jordan, Henna Koivusalo
TL;DR
The paper advances the shrinking-target problem for affine IFS by computing the exact Hausdorff dimension of the shrinking-target set $R^*(z,(\gamma^n))$ on a representative Przytycki–Urbański fractal defined by two diagonal affine maps. It develops a synthesis of dynamical methods for affine IFS with geometric analysis of Bernoulli convolutions, employing transversality and exponential-separation results to cover all centre types. It yields explicit formulas for three regimes: (i) a low-$\lambda$ regime with $\dim_H=-\log 2/\log(\lambda\gamma)$, (ii) a unique-expansion regime with $\dim_H=2+\log\lambda/\log 2-\log\gamma/\log(\gamma\lambda)$, and (iii) a typical-centre regime with $\dim_H=(2\log 2+\log\lambda)/\log(2/\gamma)$. The results bridge grid-restricted self-affine cases to the more intricate PU-fractals, highlighting the role of Bernoulli convolutions, transversality, and energy methods in dimension theory for self-affine shrinking targets. The findings contribute to understanding how shrinking targets interact with complex local geometry controlled by Bernoulli convolution measures and region-of-transversality phenomena.
Abstract
Shrinking target problems in the context of iterated function systems have received an increasing amount of interest in the past few years. The classical shrinking target problem concerns points returning infinitely many times to a sequence of shrinking balls. In the iterated function system context, the shrinking balls problem is only well tractable in the case of similarity maps, but the case of affine maps is more elusive due to many geometric-dynamical complications. In the current work, we push through these complications and compute the Hausdorff dimension of a set recurring to a shrinking target of geometric balls in some affine iterated function systems. For these results, we have pinpointed a representative class of affine iterated function systems, consisting of a pair of diagonal affine maps, that was introduced by Przytycki and Urbański. The analysis splits into many sub-cases according to the type of the centre point of the targets, and the relative sizes of the targets and the contractions of the maps, illustrating the array of challenges of going beyond affine maps with nice projections. The proofs require heavy machinery from, and expand, the theory of Bernoulli convolutions.