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Abstract We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixingcountable Markov shift with (BIP) has an entropy gap at infinity,then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistributionresults. We apply these general results to obtain counting and equidistribution results for cusped Hitchinrepresentations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.
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This content will become publicly available on February 1, 2026
Equilibrium measures for two-sided shift spaces via dimension theory
Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.
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- Award ID(s):
- 2154378
- PAR ID:
- 10618553
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 45
- Issue:
- 2
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 427 to 466
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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