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Multifractal analysis of homological growth rates for hyperbolic surfaces

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

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 , First View , pp. 1 - 35
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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

References

Aaronson, J., Denker, M. and Urbański, M.. Ergodic theory for Markov fibred systems and parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.CrossRefGoogle Scholar
Beardon, A. F.. Inequalities for certain Fuchsian groups. Acta Math. 127 (1971), 221–258.CrossRefGoogle Scholar
Beardon, A. F.. The Geometry of Discrete Groups (Graduate Texts in Mathematics, 91). Springer, New York, 1983.CrossRefGoogle Scholar
Beardon, A. F. and Maskit, B.. Limit points of Kleinian groups and finite sided fundamental polyhedra. Acta Math. 132 (1971), 1–12.CrossRefGoogle Scholar
Bowen, R.. Hausdorff dimension of quasicircles. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 11–25.CrossRefGoogle Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470), second revised edition. Springer-Verlag, Berlin, 2008.CrossRefGoogle Scholar
Bowen, R. and Series, C.. Markov maps associated with Fuchsian groups. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 153–170.CrossRefGoogle Scholar
Buzzi, J. and Sarig, O.. Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod. Th. & Dynam. Sys. 23 (2003), 1383–1400.CrossRefGoogle Scholar
Eizenberg, A., Kifer, Y. and Weiss, B.. Large deviations for ${\mathbb{Z}}^d$ -actions. Comm. Math. Phys. 164 (1994), 433–454.CrossRefGoogle Scholar
Floyd, W. J.. Group completions and limit sets of Kleinian groups. Invent. Math. 57 (1980), 205–218.CrossRefGoogle Scholar
Gelfert, K. and Rams, M.. The Lyapunov spectrum of some parabolic systems. Ergod. Th.& Dynam. Sys. 29 (2009), 919–940.CrossRefGoogle Scholar
Iommi, G.. Multifractal analysis of the Lyapunov exponent for the backward continued fraction map. Ergod. Th. & Dynam. Sys. 30 (2010), 211–232.CrossRefGoogle Scholar
Iommi, G. and Kiwi, J.. The Lyapunov spectrum is not always concave. J. Stat. Phys. 135 (2009), 535–546.CrossRefGoogle Scholar
Jaerisch, J. and Takahasi, H.. Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many branches. Adv. Math. 385 (2021), 107778.CrossRefGoogle Scholar
Johansson, A., Jordan, T., Öberg, A. and Pollicott, M.. Multifractal analysis of non-uniformly hyperbolic systems. Israel J. Math. 177 (2010), 125–144.CrossRefGoogle Scholar
Jordan, T. and Rams, M.. Multifractal analysis of weak Gibbs measures for non-uniformly expanding ${C}^1$ maps. Ergod. Th. & Dynam. Sys. 31 (2011), 143–164.CrossRefGoogle Scholar
Kesseböhmer, M. and Stratmann, B. O.. A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups. Ergod. Th. & Dynam. Sys. 24 (2004), 141–170.CrossRefGoogle Scholar
Kesseböhmer, M. and Stratmann, B. O.. A multifractal analysis for Stern–Brocot intervals, continued fractions and Diophantine growth rates. J. Reine Angew. Math. 605 (2007), 133–163.Google Scholar
Lalley, S.. Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math. 163 (1989), 1–55.CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M.. Parabolic iterated function systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1423–1447.CrossRefGoogle Scholar
Mauldin, R. D. and Urbański, M.. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge Tracts in Mathematics, 148). Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
Morita, T.. Markov systems and transfer operators associated with cofinite Fuchsian groups. Ergod. Th. & Dynam. Sys. 17 (1997), 1147–1181.CrossRefGoogle Scholar
Nakaishi, K.. Multifractal formalism for some parabolic maps. Ergod. Th. & Dynam. Sys. 20 (2000), 843–857.CrossRefGoogle Scholar
Olsen, L.. Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. J. Math. Pures Appl. (9) 82 (2003), 1591–1649.CrossRefGoogle Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136 (1976), 241–273.CrossRefGoogle Scholar
Pesin, Y.. Dimension Theory in Dynamical Systems: Contemporary Views and Applications. The University of Chicago Press, Chicago, IL, 1997.CrossRefGoogle Scholar
Pesin, Y. and Senti, S.. Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2 (2008), 397–430.CrossRefGoogle Scholar
Pesin, Y. and Weiss, H.. A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions. J. Stat. Phys. 86 (1997), 233–275.CrossRefGoogle Scholar
Pesin, Y. and Weiss, H.. The Multifractal Analysis of Birkhoff Averages and Large Deviations in Global Analysis of Dynamical Systems. Institute of Physics, Bristol, 2001, pp. 419–431.Google Scholar
Pollicott, M. and Urbański, M.. Asymptotic counting in conformal dynamical systems. Mem. Amer. Math. Soc. 271 (2021), v+139pp.Google Scholar
Pollicott, M. and Weiss, H.. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville–Pomeau transformations and applications to Diophantine approximation. Comm. Math. Phys. 207 (1999), 145–171.CrossRefGoogle Scholar
Prellberg, T. and Slawny, J.. Maps of intervals with indifferent fixed points: thermodynamic formalism and phase transitions. J. Stat. Phys. 66 (1992), 503–514.CrossRefGoogle Scholar
Ratcliffe, J. G.. Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics, 149). Springer-Verlag, New York, 1994.CrossRefGoogle Scholar
Ruelle, D.. Thermodynamic Formalism. The Mathematical Structures of Classical Equilibrium Statistical Mechanics, 2nd edn. Cambridge University Press, Cambridge, 2004.Google Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
Schmeling, J.. On the completeness of multifractal spectra. Ergod. Th. & Dynam. Sys. 19 (1999), 1595–1616.CrossRefGoogle Scholar
Series, C.. The infinite word problem and limit sets in Fuchsian groups. Ergod. Th. & Dynam. Sys. 1 (1981), 337–360.CrossRefGoogle Scholar
Series, C.. Geometrical Markov coding of geodesics on surfaces of constant negative curvature. Ergod. Th. & Dynam. Sys. 6 (1986), 601–625.CrossRefGoogle Scholar
Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.CrossRefGoogle Scholar
Urbański, M.. Parabolic Cantor sets. Fund. Math. 151 (1996), 241–277.Google Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
Weiss, H.. The Lyapunov spectrum for conformal expanding maps and Axiom-A surface diffeomorphisms. J. Stat. Phys. 95 (1999), 615–632.CrossRefGoogle Scholar
Yuri, M.. Multifractal analysis of weak Gibbs measures for intermittent systems. Comm. Math. Phys. 230 (2002), 365–388.CrossRefGoogle Scholar