arXiv subjects

Lien-Yung Kao

Publications and source records attributed to Lien-Yung Kao.

11 recordsLinked to original sources

arXiv2025

Correlation Number for Potentials with Entropy Gaps and Cusped Hitchin Representations

We introduce a correlation number for two strictly positive, locally Hölder continuous, independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with BIP. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

math.DS↗

arXiv2025

Pressure gaps, geometric potentials, and nonpositively curved manifolds

In this paper, we derive a general pressure gap criterion for closed rank 1 manifolds whose singular sets are given by codimension 1 totally geodesic flat subtori. As an application, we show that under certain curvature constraints, potentials that decay faster than geometric potentials (towards to the singular set) have pressure gaps and have no phase transitions. Along the way, we prove that geometric potentials are Hölder continuous near singular sets.

math.DS↗

arXiv2023

Pressure metrics for cusped Hitchin components

We study the cusped Hitchin component consisting of (conjugacy classes of) cusped Hitchin representations of a torsion-free geometrically finite Fuchsian group into PSL(d,R). We produce pressure metrics associated to the first fundamental weight and the first simple root. When $d=3$ we produce a pressure metric associated to the Hilbert length, which is new even when the Fuchsian group is cocompact.

math.GT↗

arXiv2022

Counting, equidistribution and entropy gaps at infinity with applications to cusped Hitchin representations

We show that if an eventually positive, non-arithmetic, locally Hölder continuous potential for a topologically mixing countable Markov shift with (BIP) has an entropy gap at infinity, then one may apply the renewal theorem of Kesseböhmer and Kombrink to obtain counting and equidistribution results. We apply these general results to obtain counting and equidistribution results for cusped Hitchin representations, and more generally for cusped Anosov representations of geometrically finite Fuchsian groups.

math.DS↗

arXiv2021

Pressure metrics for deformation spaces of quasifuchsian groups with parabolics

In this paper, we produce a mapping class group invariant pressure metric on the space QF(S) of quasiconformal deformations of a co-finite area Fuchsian group uniformizing a surface S. Our pressure metric arises from an analytic pressure form on QF(S) which is degenerate only on pure bending vectors on the Fuchsian locus. Our techniques also show that the Hausdorff dimension of the limit set varies analytically over QF(S).

math.GT↗

arXiv2019

Properties of equilibrium states for geodesic flows over manifolds without focal points

We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.

math.DS↗

arXiv2019

Pressure metrics and Manhattan curves for Teichmüller spaces of punctured surfaces

In this paper, we extend the construction of pressure metrics to Teichmüller spaces of surfaces with punctures. This construction recovers Thurston's Riemannian metric on Teichmüller spaces. Moreover, we prove the real analyticity and the convexity of Manhattan curves of the finite area type-preserving Fuchsian representations, and thus we obtain several related entropy rigidity results. Lastly, relating the two topics mentioned above, we show that one can derive the pressure metric by varying Manhattan curves.

math.DS↗

arXiv2018

Unique equilibrium states for geodesic flows over surfaces without focal points

In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states. We show these unique equilibrium states are Bernoulli, and weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.

math.DS↗

arXiv2018

Manhattan Curves for Hyperbolic Surfaces with Cusps

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 and Richard Sharp for convex-cocompact Fuchsian representations.

math.DS↗

arXiv2016

Pressure type metrics on spaces of metric graphs

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichmüller theory. This paper is motivated by Pollicott and Sharp's earlier work.

math.DS↗

arXiv2015

Entropy, Critical Exponent and Immersed Surfaces in Hyperbolic 3-Manifolds

We consider a $π_{1}$--injective immersion $f:Σ\to M$ from a compact surface $Σ$ to a hyperbolic 3--manifold $M$. Let $Γ$ denote the copy of $π_{1}Σ$ in $\mathrm{Isom}({\mathbb{H}}^{3})$ induced by the immersion and $δ(Γ)$ be the critical exponent. Suppose $Γ$ is convex cocompact and $Σ$ is negatively curved, we prove that there are two geometric constants $C_{1}(Σ,M)$ and $C_{2}(Σ,M)$ not bigger than $1$ such that $C_{1}(Σ,M)\cdotδ_Γ\leq h(Σ)\leq C_{2}(Σ,M)\cdotδ_Γ$, where $h(Σ)$ is the topological entropy of the geodesic flow on. When $f$ is an embedding, we show that $C_{1}(Σ,M)$ and $C_{2}(Σ,M)$ are exactly the geodesic stretches (a.k.a. Thurston's intersection number) with respect to certain Gibbs measures. Moreover, we prove the rigidity phenomenon arising from this inequality. Lastly, as an application, we discuss immersed minimal surfaces in hyperbolic 3--manifolds and these discussions lead us to results similar to A. Sanders' work on the moduli space of $Σ$ introduced by C. Taubes.

math.DS↗