Related papers: Ruelle's zeta function for non-Archimedean rationa…
Let $K$ be a complete non-archimedean field of characteristic $0$ equipped with a discrete valuation. We establish the rationality of the Artin-Mazur zeta function on the Julia set for any subhyperbolic rational map defined over $K$ with a…
In this article we introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers…
In this paper we consider the transfer operator approach to the Ruelle and Selberg zeta functions associated to continued fractions transformations and the geodesic flow on the full modular surface. We extend the results by Ruelle and Mayer…
In this article, we study local zeta functions over non-Archimedean locals fields of arbitrary characteristic attached to rational functions and characters $\chi$ of the units of the ring of integers $\mathcal{O}_{K}$, by using an approach…
We construct the first examples of rational functions defined over a non-archimedean field with certain dynamical properties. In particular, we find such functions whose Julia sets, in the Berkovich projective line, are connected but not…
We consider applications of transfer operators (also known as Ruelle operators) to completely positive maps (CPT) in quantum information theory. It is described a correspondence between fixed points of CPT maps and certain Markov-invariant…
This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by $\exp(C_K |s|^{\delta})$ in strips $|\Re s| \leq K$, where…
In this article, we introduce a notion of non-degeneracy, with respect to certain Newton polyhedra, for rational functions over non-Archimedean locals fields of arbitrary characteristic. We study the local zeta functions attached to…
We study relations between non-commutative Ruelle transfer operators over the C$^*$-algebra $B(\mathcal{H})$ of linear bounded operators over separable Hilbert spaces $\mathcal{H}$ (infinite-dimensional) and other completely positive maps.…
We prove that, for a Ruelle-expanding map, the zeta function is rational and the topological entropy is equal to the exponential growth rate of the periodic points.
We present a numerical algorithm for the computation of invariant Ruelle distributions on convex co-compact hyperbolic surfaces. This is achieved by exploiting the connection between invariant Ruelle distributions and residues of…
Consider a rational map $R$ of degree $d\geq 2$ with coefficients over the non-archimedean field $\mathbb{C}_p$, with $p$ a fixed prime number. If $R$ has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier…
We will prove that the zeta function for Ruelle-expanding maps is rational.
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…
If $f$ is a transcendental entire function with only algebraic singularities we calculate the Ruelle operator of $f$. Moreover, we prove both (i) if $f$ has a summable critical point, then $f$ is not structurally stable under certain…
We exhibit an analytic family of hyperbolic, even disjoint type, entire functions for which the hyperbolic dimension does not vary analytically. Additionally we answer several questions in thermodynamic formalism of entire functions such as…
Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $\pi = \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $\pi = \mathbb{Z}$, the $L^2$ zeta…
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire…
By building on former results and the cusp expansion algorithm, we construct strict transfer operator approaches for geometrically finite developable hyperbolic orbisurfaces of infinite area without cusps. Together with the cusp expansion…
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…