Localized asymptotic behavior for almost additive potentials
Abstract
We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of {\it weak} concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of $\phi_n(x)$ is localized, i.e. depends on the point $x$ rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form $\{x\in X: \lim_{n\to\infty} \phi_n(x)/n=\xi(x)\}$, where $\xi$ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in $\R^d$, as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2011
- DOI:
- 10.48550/arXiv.1104.1442
- arXiv:
- arXiv:1104.1442
- Bibcode:
- 2011arXiv1104.1442B
- Keywords:
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- Mathematics - Dynamical Systems;
- Primary: 37B40;
- Secondary: 28A80