What the study found
The authors compute the Hausdorff dimension of a set of points that recur infinitely often to a shrinking target of geometric balls in certain affine iterated function systems. Their results apply to a representative class of affine systems made of a pair of diagonal affine maps.
Why the authors say this matters
The study suggests that shrinking target problems in affine iterated function systems can be handled despite the geometric and dynamical complications that make them harder than the similarity-map case. The authors also indicate that their analysis shows the range of challenges involved in moving beyond affine maps with nice projections.
What the researchers tested
The researchers studied shrinking target problems in iterated function systems, a setting where points return repeatedly to a sequence of shrinking balls. They focused on affine maps, and specifically on a class introduced by Przytycki and Urbański consisting of two diagonal affine maps. Their analysis splits into multiple sub-cases depending on the target center point and on the relative sizes of the targets and the map contractions.
What worked and what didn't
The study reports that the Hausdorff dimension was computed for the recurring set in the affine systems considered. The analysis required heavy machinery from the theory of Bernoulli convolutions, and the authors say it expands that theory. The abstract does not describe any failed cases or negative results.
What to keep in mind
The results are stated only for some affine iterated function systems, not for all such systems. The abstract notes that the analysis depends on several sub-cases and on the geometry of the target center and target size; it does not give more detailed limitations in the available summary.
Key points
- The paper computes the Hausdorff dimension of a shrinking-target recurrence set.
- The setting is an affine iterated function system with a pair of diagonal affine maps.
- The analysis divides into sub-cases based on the target center and the relative sizes of targets and contractions.
- The proofs use heavy machinery from the theory of Bernoulli convolutions.
- The abstract does not describe any failed cases or broader limitations beyond the specific class studied.
Disclosure
- Research title:
- Hausdorff dimension computed for shrinking targets in affine systems
- Authors:
- Thomas Jordan, Henna Koivusalo
- Publication date:
- 2026-04-22
- OpenAlex record:
- View
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