Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps

Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations

Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

Symbolic Dynamics for the Geodesic Flow on Two-dimensional Hyperbolic Good Orbifolds

We construct cross sections for the geodesic flow on the orbifolds $Γ\backslash H$ which are tailor-made for the requirements of transfer operator approaches to Maass cusp forms and Selberg zeta functions. Here, $H$ denotes the hyperbolic plane and $Γ$ is a nonuniform geometrically finite Fuchsian group (not necessarily a lattice, not necessarily arithmetic) which satisfies an additional condition of geometric nature. The construction of the cross sections is uniform, geometric, explicit and algorithmic.

通过有限型移位和ζ函数研究二次易系数猜想

The Quadratic Easy Coefficients Conjecture via Finite-Type Shifts and Zeta Functions

本文通过有限型移位和ζ函数，证明了二次易系数猜想，推导了易系数公式，并论证了权重递推的非奇异性与唯一反向延拓性。

Ihara Zeta Entropy

In this article, we introduce an entropy based on the formal power series expansion of the Ihara Zeta function. We find a number of inequalities based on the values of the Ihara zeta function. These new entropies are applicable in symbolic dynamics and the dynamics of billiards.

关于符号动力学的某些带标签有向图

On certain labelled directed graphs of symbolic dynamics

本文针对包含 Dyck 位移的编码系统族，提出次同步化与受限复杂度概念，并显式构造扩展 Dyck 位移的次移位族，证明其可由受限复杂度和 Artin-Mazur zeta 函数等拓扑共轭不变量共同刻画。