What the study found
The study computes the Hausdorff dimension of a set of points that recur to shrinking geometric targets in certain affine iterated function systems. The authors focus on a representative class of diagonal affine maps introduced by Przytycki and Urbański.
Why the authors say this matters
The authors say this work pushes through the complications that make shrinking-target problems in affine systems difficult. They also conclude that the analysis illustrates the challenges of moving beyond affine maps with nice projections and expands the theory of Bernoulli convolutions.
What the researchers tested
The researchers studied shrinking target problems in the setting of iterated function systems, where points return infinitely many times to a sequence of shrinking balls. They examined affine maps rather than similarity maps, and their analysis split into multiple sub-cases depending on the target center and the relative sizes of the targets and the contractions of the maps.
What worked and what didn't
The paper reports that the Hausdorff dimension can be computed for the chosen affine system and shrinking-target setting. It also states that the problem is more tractable for similarity maps, while the affine case is more elusive because of geometric and dynamical complications.
What to keep in mind
The abstract does not give the detailed formula for the dimension or list the sub-case outcomes. It also does not describe limitations beyond noting that the results apply to a specific class of diagonal affine iterated function systems.
Key points
- The study computes the Hausdorff dimension of a shrinking-target set for certain affine iterated function systems.
- The systems considered are a pair of diagonal affine maps introduced by Przytycki and Urbański.
- The analysis depends on the center of the target point and the relative sizes of targets and map contractions.
- The abstract says affine shrinking-target problems are harder than the similarity-map case.
- The proofs use and extend theory from Bernoulli convolutions.
Disclosure
- Research title:
- Hausdorff dimension computed for shrinking targets on affine fractals
- Authors:
- Thomas Jordan, Henna Koivusalo
- Publication date:
- 2026-04-22
- DOI:
- 10.1090/tran/9799
- OpenAlex record:
- View
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