Related papers: Ruelle's zeta function for non-Archimedean rationa…

For geometrically finite non-compact developable hyperbolic orbisurfaces (including those of infinite volume), we provide transfer operator families whose Fredholm determinants are identical to the Selberg zeta function. Our proof yields an…

Let $f:\hat{\mathbb C}\to\hat{\mathbb C}$ be a hyperbolic rational map of degree $d\ge2$ on the Riemann sphere. We give several conditions which are equivalent to the condition for the Julia set $J_f$ to be a Cantor set. It has been known…

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then…

Let $K$ denote either $\mathbb{R}$ or $\mathbb{C}$. In this article, we introduce two new equivariants associated to a rational map $f\in K(z)$. These objects naturally live on a real hyperbolic space, and carry information about the action…

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 $(V,\chi)$ for hyperbolic surfaces…

We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for automorphisms of non-finitely generated torsion abelian groups and for endomorphisms of groups $\mathbb Z_p^d,$…

We show that for any $\lambda \in \mathbb{C}$ with $|\lambda|<1$ there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative…

Generalizing Emerton's completed cohomologies, we define relative completed cohomologies of arithmetic manifolds. We also define modular symbols for them, and show that the relative completed cohomology spaces interpolate the ``nearly…

We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each…

This is an integrated part of our Geo-Arithmetic Program. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields by a weighted count of semi-stable bundles. Basic…

Let $B$ be a rational function of degree at least two that is neither a Latt\`es map nor conjugate to $z^{\pm n}$ or $\pm T_n$. We provide a method for describing the set $C_B$ consisting of all rational functions commuting with $B.$…

We introduce the notion of a regular mapping on a non-commutative $L_p$-space associated to a hyperfinite von Neumann algebra for $1\le p\le \infty$. This is a non-commutative generalization of the notion of regular or order bounded map on…

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle…

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis…

Let $L$ be a solvable Lie algebra of dimension less than or equal to 4 over finite fields. We compute and record, in explicit symbolic form, the zeta functions enumerating subalgebras or ideals of $L$, and study their properties. We also…

In this article we prove an important inequality regarding the Ruelle operator in hyperbolic flows. This was already proven briefly by Mark Pollicott and Richard Sharp in a low dimensional case, but we present a clearer proof of the…

For a reductive group G over a non-archimedean local field, we compare smooth representations over C with smooth representations over Qbar (an algebraic closure of Q). We show that an elliptic G-representation (in the sense of Arthur) can…

The characteristic function has been an important tool for studying completely non unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space…

Let \(\mathcal{A}\) be a finite-dimensional real (or complex) C*-algebra, \(\Omega_{A}\) an aperiodic subshift of finite type, and \(\mathcal{C}(\Omega_{A}; \mathcal{A})\) the set of continuous functions from \(\Omega_{A}\) to…

We describe a general method to prove meromorphic continuation of dynamical zeta functions to the entire complex plane under the condition that the corresponding partition functions are given via a dynamical trace formula from a family of…