Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

The Taylor expansion of Ruelle L-function at the origin and the Borel regulator

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rational multiple of the volume up to a certain contribution from cusps. Moreover if the dimension is three we will identify the leading coefficient. Both of them will be intepreted by the Borel regulator in algebraic K-theory. Also a relation with the L^2-torsion will be discussed.

扭曲的动力学ζ函数与弗里德猜想

Twisted dynamical zeta functions and the Fried's conjecture

本文综述了鲁埃勒和塞伯格的扭曲动力学ζ函数及弗里德猜想，基于作者在 Institut Henri Poincaré 的迷你课程内容。

Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points

We consider the Ruelle zeta function $R(s)$ of a genus $g$ hyperbolic Riemann surface with $n$ punctures and $v$ ramification points. $R(s)$ is equal to $Z(s)/Z(s+1)$, where $Z(s)$ is the Selberg zeta function. The main result of this work is the leading behavior of $R(s)$ at $s=0$. If $n_0$ is the order of the determinant of the scattering matrix $φ(s)$ at $s=0$, we find that \begin{align*} \lim_{s\rightarrow 0}\frac{R(s)}{s^{2g-2+n-n_0}}=(-1)^{\frac{A}{2}+1}(2π)^{2g-2+n }\tildeφ(0)^{-1} \prod_{j=1}^v m_j, \end{align*}which says that $R(s)$ has order $2g-2+n-n_0$ at $s=0$, and its leading coe…

关于扭曲Ruelle zeta函数的函数方程及Fried猜想

On the functional equation of twisted Ruelle zeta function and Fried's conjecture

本文证明了扭曲Ruelle zeta函数满足的函数方程，并利用其确定零点阶数与Laurent展开首项，结合已有结果推进了Fried猜想的证明。

A dynamical systems analogue of Lichtenbaum's conjectures on special values of Hasse-Weil zeta functions

In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-étale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of Álvarez López and Kordyukov we express the leading coefficient $ζ^*_R (0)$ of the Ruelle zeta function $ζ_R (s)$ at $s = 0$ in terms of analyt…

Fuglede-Kadison determinants over free groups and Lehmer's constants

Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap. In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound $\frac{2}{\sqrt{3}}$ for Lehmer's constants of all torsion-free groups which have non-cyclic free subgroups. Our proofs use relations between Fuglede-Kadison determinants and random…