What the study found
The study found that any recurrence set of a subshift with the authors’ (W′)-specification property has full Hausdorff dimension, meaning it is as large as possible in a fractal-dimension sense. The result also extends to a wide class of subshifts without specification and to certain piecewise expanding interval maps.
Why the authors say this matters
The authors suggest this matters because it extends full Hausdorff dimension results beyond systems with specification, a strong regularity property in dynamical systems. They also state that the result applies to a wide class of systems, including S-gap shifts, some coded shifts, and coding spaces of transitive piecewise monotonic interval maps with positive entropy.
What the researchers tested
The researchers introduced (W′)-specification in terms of language decompositions of subshifts. They then studied recurrence sets in subshifts and in piecewise expanding interval maps.
What worked and what didn't
For subshifts with (W′)-specification, the recurrence sets were shown to have full Hausdorff dimension. The same conclusion was obtained for a wide class of subshifts without specification, including all S-gap shifts, some coded shifts, and coding spaces of transitive piecewise monotonic interval maps with positive entropy; for a wide class of piecewise expanding interval maps, any recurrence set has full Hausdorff dimension.
What to keep in mind
The abstract does not describe limitations, exceptions, or cases where the result fails. It also does not give the detailed definitions of recurrence sets, Hausdorff dimension, or the (W′)-specification property beyond the brief description in the abstract.
Key points
- The paper shows full Hausdorff dimension for recurrence sets in subshifts with (W′)-specification.
- The result extends to a wide class of subshifts without specification.
- Examples named in the abstract include all S-gap shifts and some coded shifts.
- The result also applies to coding spaces of transitive piecewise monotonic interval maps with positive entropy.
- For a wide class of piecewise expanding interval maps, any recurrence set has full Hausdorff dimension.
Disclosure
- Research title:
- Recurrence sets have full Hausdorff dimension in several subshifts
- Authors:
- Hiroki Takahasi
- Institutions:
- Keio University
- Publication date:
- 2026-04-22
- OpenAlex record:
- View
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