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Multifractal analysis of homological growth rates for hyperbolic surfaces

Part of: Dynamical systems with hyperbolic behavior Low-dimensional dynamical systems Smooth dynamical systems: general theory

Published online by Cambridge University Press:  18 September 2024

JOHANNES JAERISCH  and
HIROKI TAKAHASI
JOHANNES JAERISCH*
Affiliation:
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
HIROKI TAKAHASI
Affiliation:
Keio Institute of Pure and Applied Sciences (KiPAS), Department of Mathematics, Keio University, Yokohama 223-8522, Japan (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

Keywords

Fuchsian groupBowen–Series mapthermodynamic formalismmultifractal analysis

MSC classification

Primary: 37C45: Dimension theory of dynamical systems 37D35: Thermodynamic formalism, variational principles, equilibrium states 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 3 , March 2025 , pp. 849 - 883
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

Dedicated to Professor Masato Tsujii on the occasion of his 60th birthday

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