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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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