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Manhattan curves for hyperbolic surfaces with cusps

Part of: Riemann surfaces Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  04 December 2018

LIEN-YUNG KAO Open the ORCID record for LIEN-YUNG KAO [Opens in a new window]
LIEN-YUNG KAO*
Affiliation:
Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA email [email protected]
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Abstract

In this paper, we study an interesting curve, the so-called Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface; in particular, representations corresponding to Riemann surfaces with cusps. Using thermodynamic formalism (for countable state Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Burger [Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not.1993(7) (1993), 217–225] and Sharp [The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z.228(4) (1998), 745–750] for convex cocompact Fuchsian representations.

Keywords

group actionslow dimensional dynamicssmooth dynamicssymbolic dynamics

MSC classification

Primary: 37D35: Thermodynamic formalism, variational principles, equilibrium states
Secondary: 30F60: Teichmüller theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1843 - 1874
Copyright
© Cambridge University Press, 2018

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References

Ahlfors, L. V.. Lectures on Quasiconformal Mappings (University Lecture Series, 38), 2nd edn. American Mathematical Society, Providence, RI, 2006; with supplementary chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard.Google Scholar
Barreira, L. and Iommi, G.. Suspension flows over countable Markov shifts. J. Stat. Phys. 124(1) (2006), 207–230.Google Scholar
Bishop, C. and Steger, T.. Representation-theoretic rigidity in PSL (2, R). Acta Math. 170(1) (1993), 121–149.Google Scholar
Burger, M.. Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank 2. Int. Math. Res. Not. IMRN 7 (1993), 217–225.Google Scholar
Dal’bo, F. and Kim, I.. Shadow lemma on the product of Hadamard manifolds and applications. Actes du Séminaire de Théorie Spectrale et Géométrie (Séminaire de Théorie Spectrale et Géométrie, 25). Université Grenoble I, Saint-Martin-d’Hères, 2008, pp. 105–119, Année 2006–2007.Google Scholar
Dal’bo, F. and Peigné, M.. Groupes du ping-pong et géodésiques fermées en courbure - 1. Ann. Inst. Fourier (Grenoble) 46(3) (1996), 755–799.Google Scholar
Dal’bo, F. and Peigné, M.. Some negatively curved manifolds with cusps, mixing and counting. J. Reine Angew. Math. 497(1617430) (1998), 141–169.Google Scholar
Iommi, G., Riquelme, F. and Velozo, A.. Entropy in the cusp and phase transitions for geodesic flows. Israel J. Math. 225(2) (2018), 609–659.Google Scholar
Jaerisch, J., Kesseböhmer, M. and Lamei, S.. Induced topological pressure for countable state Markov shifts. Stoch. Dyn. 14(2) (2014), 1350016–1350031.Google Scholar
Kapovich, M.. Hyperbolic manifolds and discrete groups. Modern Birkhäuser Classics. Birkhäuser Boston, Boston, MA, 2009.Google Scholar
Kempton, T.. Thermodynamic formalism for suspension flows over countable Markov shifts. Nonlinearity 24(10) (2011), 2763–2775.Google Scholar
Kim, I.. Marked length rigidity of rank one symmetric spaces and their product. Topology 40(6) (2001), 1295–1323.Google Scholar
Maskit, B.. Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Vol. 287. Springer, Berlin, 1988.Google Scholar
Daniel Mauldin, R. and Urbański, M.. Gibbs states on the symbolic space over an infinite alphabet. Israel J. Math. 125(1) (2001), 93–130.Google Scholar
Daniel Mauldin, R. and Urbański, M.. Graph Directed Markov Systems (Cambridge Tracts in Mathematics, 148). Cambridge University Press, Cambridge, 2003.Google Scholar
Otal, J.-P. and Peigné, M.. Principe variationnel et groupes kleiniens. Duke Math. J. 125(1) (2004), 15–44.Google Scholar
Peigné, M.. Autour de l’exposant de Poincaré d’un groupe kleinien. Géométrie ergodique (Enseignement Mathématique Monograph, 43). L’Enseignement Mathématique, Geneva, 2013, pp. 25–59.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Astérisque (2015), 373.Google Scholar
Roblin, T.. Ergodicité et équidistribution en courbure négative. Mém. Soc. Math. Fr. (N.S.) 95 (2003), vi+96.Google Scholar
Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19(6) (1999), 1565–1593.Google Scholar
Sarig, O.. Phase transitions for countable Markov shifts. Comm. Math. Phys. 217(3) (2001), 555–577.Google Scholar
Sarig, O.. Existence of Gibbs measures for countable Markov shifts. Proc. Amer. Math. Soc. 131(6) (2003), 1751–1758 (electronic).Google Scholar
Sarig, O.. Lecture notes on thermodynamic formalism for topological markov shifts, Preprint, 2009, available at http://www.weizmann.ac.il/math/sarigo/sites/math.sarigo/files/uploads/tdfnotes.pdf.Google Scholar
Savchenko, S. V.. Special flows constructed from countable topological Markov chains. Funktsional. Anal. i Prilozhen. 32(1) (1998), 40–53, 96.Google Scholar
Schapira, B.. Lemme de l’ombre et non divergence des horosphères d’une variété géométriquement finie. Ann. Inst. Fourier (Grenoble) 54(4) (2004), 939–987.Google Scholar
Sharp, R.. The Manhattan curve and the correlation of length spectra on hyperbolic surfaces. Math. Z. 228(4) (1998), 745–750.Google Scholar
Thurston, W. P.. Minimal stretch maps between hyperbolic surfaces. Preprint, 1998, https://arxiv.org/pdf/math/9801039.pdf.Google Scholar
Tukia, P.. On isomorphisms of geometrically finite Möbius groups. Publ. Math. Inst. Hautes Études Sci. 61(1) (1985), 171–214.Google Scholar