Related papers: Dynamical zeta functions for analytic surface diff…

We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of…

I show that the dynamical determinant, associated to an Anosov diffeomorphism, is the Fredholm determinant of the corresponding Ruelle-Perron-Frobenius transfer operator acting on appropriate Banach spaces. As a consequence it follows, for…

This paper explores the domain of meromorphic extension for the dynamical zeta function associated to a class of one-dimensional differentiable parabolic maps featuring an indifferent fixed point. We establish the connection between this…

We study regularized determinants of Laplacians acting on the space of Hilbert-Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in…

Let $s_0 < 0$ be the abscissa of absolute convergence of the dynamical zeta function $Z(s)$ for several disjoint strictly convex compact obstacles $K_i \subset \R^N, i = 1,..., \kappa_0,\: \ka_0 \geq 3,$ and let $R_{\chi}(z) = \chi…

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds…

This is a survey article on the twisted dynamical zeta functions of Ruelle and Selberg and the Fried's conjecture. It is based on the mini-course: "Twisted Ruelle zeta function, complex-valued analytic torsion and the Fried's conjecture",…

In this paper, we consider the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd dimensional manifold $X$. These functions are initially defined on one complex variable $s$ in some…

Let $ R $ be a rational map. We are interesting in the dynamic of the Ruelle operator on suitable spaces of differentials. In particular the necessary and sufficient conditions (in terms of convergence of sequences of measures) of existence…

We study the number of the poles of the meromorphic continuation of the dynamical zeta functions $\eta_N$ and $\eta_D$ for several strictly convex disjoint obstacles satisfying non-eclipse condition. We obtain a strip $\{z \in \mathbb C:\:…

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or…

Dynamical zeta functions are expected to relate the Schr\"odinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationsship is exact in the case of surfaces of constant negative…

For proper group actions on smooth manifolds, with compact quotients, we define an equivariant version of the Ruelle dynamical $\zeta$-function for equivariant flows satisfying a nondegeneracy condition. The construction is based on an…

A Hermite type formula is introduced and used to study the zeta function over the real and complex n-projective space. This approach allows to compute the residua at the poles and the value at the origin as well as the value of the…

Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in {\mathbb C},\: {\rm Im}\: \mu_j > 0,$ be the resonances of the Laplacian in the…

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the…

We show an equality between the analytic torsion and the absolute value at the zero point of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the…

We study invariants of a plane cuve singularity $(f,0)$ coming from motivic integration on symmetric powers of a formal deformation of $f$. We show that a natural discriminant integral recovers the motivic classes of the principal Hilbert…

We derive simple new expressions, in various dimensions, for the functional determinant of a radially separable partial differential operator, thereby generalizing the one-dimensional result of Gel'fand and Yaglom to higher dimensions. We…