Manhattan curves for hyperbolic surfaces with cusps

Abstract

In this paper, we study an interesting curve, so-called the Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especially representations corresponding to Riemann surfaces with cusps. Using Thermodynamic Formalism (for countable Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Marc Burger [Bur93] and Richard Sharp [Sha98] for convex-cocompact Fuchsian representations.

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

“…To derive the analyticity of pressure, we need to locate the place where phase transition happens. As in [Kao18], we have the following observation.…”

“…In this section, we will prove Theorem A and Theorem B. The ideas mostly follow [Kao18]. In [Kao18], the author used results of Paulin, Pollicott and Schapira [PPS15] to analyze the geometric Gurevich pressure over the geodesics flow.…”

“…The ideas mostly follow [Kao18]. In [Kao18], the author used results of Paulin, Pollicott and Schapira [PPS15] to analyze the geometric Gurevich pressure over the geodesics flow. The general framework in [PPS15] includes finite area hyperbolic surfaces.…”

“…The general framework in [PPS15] includes finite area hyperbolic surfaces. Nevertheless, for completeness, we will give outlines of the proofs, and the reader can find all the details in [Kao18].…”

“…Using similar ideas to those in [LS08,Kao18], we study geodesic flows over hyperbolic surfaces with cusps by countable state Markov shifts and corresponding suspension flows. Notice that for countable state Markov shifts, in contrast to compact cases, for unbounded potentials without sufficient control of their regularity and values around cusps, the pressure of their perturbation might not only lose the analyticity but also information of some thermodynamics data.…”