What the study found
The authors introduce a property called (W′)-specification for subshifts, which are symbolic dynamical systems, and show that any recurrence set of a subshift with this property has full Hausdorff dimension. They also report that this applies to a wide class of subshifts without specification, including all S-gap shifts, some coded shifts, and the coding space of any transitive piecewise monotonic interval map with positive entropy.
What the authors say this matters
The abstract says the result applies beyond systems with specification, so the study suggests the full-dimension conclusion holds in a broader setting than previously covered by specification-based results. The findings indicate an additional class of dynamical systems for which recurrence sets have full Hausdorff dimension.
What the researchers tested
The study develops (W′)-specification in terms of language decompositions of subshifts. The authors then use this framework to study recurrence sets in subshifts and, further, in a wide class of piecewise expanding interval maps.
What worked and what didn't
For subshifts with (W′)-specification, the authors show recurrence sets have full Hausdorff dimension. The main result also applies to several systems 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, they show any recurrence set has full Hausdorff dimension.
What to keep in mind
The abstract does not describe limitations, exceptions, or negative results beyond the scope of the classes named. All claims here are limited to the systems explicitly mentioned in the abstract.
Key points
- The paper introduces (W′)-specification for subshifts.
- For subshifts with (W′)-specification, recurrence sets have full Hausdorff dimension.
- The result is said to apply to systems without specification, including all S-gap shifts and some coded shifts.
- The coding space of any transitive piecewise monotonic interval map with positive entropy is included in the stated scope.
- 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 beyond specification
- Authors:
- Hiroki Takahasi
- Institutions:
- Keio University
- Publication date:
- 2026-04-22
- OpenAlex record:
- View
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