Dynamical Zeta Function Research Articles

The Determinant of the Dirichlet-to-Neumann Map for Surfaces with Boundary

For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of main value at 0 of a Ruelle zeta function using uniformization of Mazzeo-Taylor. We apply it to compact hyperbolic surfaces with totally geodesic boundary and obtain various interpretations of the determinant of the Dirichlet Laplacian in terms of dynamical zeta functions. We also relate in any dimension the DN map for the Yamabe operator to the scattering operator for a conformally compact related problem by using uniformization.

Unstable periodic orbits in weak turbulence

We report on a sophisticated numerical study of a parallel space–time algorithm for the computation of periodic solutions of the driven, incompressible Navier–Stokes equations in the turbulent regime. Efforts to apply the machinery of dynamical systems theory to fluid turbulence depend on the ability to accurately and reliably compute such unstable periodic orbits (UPOs). For example, the UPOs may be used to construct the dynamical zeta function of the system, from which very accurate turbulent averages of observables may be extracted. Though a number of algorithms for computing such orbits have been proposed and tested, in this paper we focus on a space–time variational principle introduced by Lan and Cvitanović in 2004 [15]. This method has not, to our knowledge, been tested on dynamical systems of high dimension because of the formidable storage and computation required. In this paper, we use petascale computation to apply this algorithm to weak hydrodynamic turbulence. We begin with a brief description and reformulation of the space–time algorithm of Lan and Cvitanović. We then describe how to apply this algorithm to the lattice-Boltzmann method for the solution of the Navier–Stokes equations. In particular, we describe the fully parallel implementation of this algorithm using the Message Passing Interface. This implementation, called HYPO4D, has been successfully deployed on a large variety of platforms both in the UK and the USA and has shown very good scalability to tens of thousands of computing cores. Finally, we describe the application of this implementation to the problem of weak homogeneous turbulence driven by an Arnold–Beltrami–Childress force field in three spatial dimensions, at a Reynolds number of 371. We commence by systematically searching for nearly periodic orbits as candidate solutions from which to begin the relaxation; we then apply the variational algorithm until convergence is obtained. Because the algorithm requires storage of the space–time lattice, even the smallest orbits require resources on the order of tens of thousands of computing cores. Using this approach, two UPOs have been identified and some of their properties have been analysed.

Scattering Phase Correction for Semiclassical Quantization Rules in Multi-Dimensional Quantum Systems

While the scattering phase for several one-dimensional potentials can be exactly derived, less is known in multi-dimensional quantum systems. This work provides a method to extend the one-dimensional phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamical zeta functions, and semiclassical Landauer–Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.

Dynamics of asymptotically hyperbolic manifolds

We prove a dynamical wave trace formula for asymptotically hyperbolic (n + 1)-dimensional manifolds with negative (but not necessarily constant) sectional curvatures; the formula equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duistermaat and Guillemin for compact manifolds and the results of Guillope and Zworski, Perry, and Guillarmou and Naud for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex cocompact hyperbolic manifolds. We define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a prime orbit theorem for the geodesic flow. This is the first such result for manifolds that have neither constant curvature nor finite volume. As a corollary to the prime orbit theorem, using our dynamical resonance-wave trace formula, we show that the existence of pure point spectrum for the Laplacian on negatively curved compact perturbations of convex cocompact hyperbolic manifolds is related to the dynamics of the geodesic flow.

A note on the dynamical zeta function of general toral endomorphisms

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.

On a symbolic dynamical zeta function and applications

A symbolic dynamical zeta function for subshifts over a countable alphabet is analyzed. We present examples on interval maps and suspensions.

Local and global statistical dynamical properties of chaotic Markov analytic maps and repellers: A coarse grained and spectral perspective

The statistical properties of chaotic Markov analytic maps and equivalent repellers are investigated through matrix representations of the Frobenius–Perron operator ( U ). The associated basis sets are constructed using Chebyshev functions and Markov partitions which can be tailored to examine statistical dynamical properties associated with observables having support over local regions or for example, about periodic orbits. The decay properties of corresponding time correlations functions are given by a analytic expression of the spectra of U which is expected to be valid for a much larger class of systems than that studied here. An explicit and general expression is also derived for the correction factor to the dynamical zeta functions occurring when analytic function spaces are not invariant under U .

Periodic orbits of linear endomorphisms on the 2-torus and its lattices

Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism.

Spatial complexity in multi-layer cellular neural networks

This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.

A Dynamical Zeta Function for Pseudo Riemannian Foliations

We investigate a generalization of geodesic random walks to pseudo Riemannian foliations. The main application we have in mind is to consider the logarithm of the associated zeta function as grand canonical partition function in a theory unifying aspects of general relativity, quantum mechanics and dynamical systems.

Dynamical Zeta Function for Several Strictly Convex Obstacles

AbstractThe behavior of the dynamical zeta function ZD(s) related to several strictly convex disjoint obstacles is similar to that of the inverse Q(s) = of the Riemann zeta function ζ(s). Let Π(s) be the series obtained from ZD(s) summing only over primitive periodic rays. In this paper we examine the analytic singularities of ZD(s) and Π(s) close to the line , where s2 is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions ZD(s), Π(s) has a singularity at s = s2.

Spectra of Perron–Frobenius operator and new construction of two dimensional low discrepancy sequences

For high dimensional transformations, the essential spectral radius of the Perron–Frobenius operator restricted to some natural space is generally smaller than the reciprocals of the radius of convergence of the dynamical zeta function. We will construct transformations whose essential spectral radius coincide with the reciprocal of the radius of convergence of the dynamical zeta functions, and construct new type of 2 dimensional low discrepancy sequences.

Meromorphic continuation of dynamical zeta functions via transfer operators

We describe a 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 transfer operators. Further we give general conditions for the partition functions associated with general spin chains to be of this type and provide various families of examples for which these conditions are satisfied.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

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 ⊂ R N , i = 1 , … , κ 0 , κ 0 ⩾ 3 and let R χ ( z ) = χ ( − Δ D − z 2 ) −1 χ , χ ∈ C 0 ∞ ( R N ) , be the cut-off resolvent of the Dirichlet Laplacian − Δ D in Ω = R N ∖ ⋃ i = 1 κ 0 K i ¯ . We prove that there exists σ 2 < s 0 such that Z ( s ) is analytic for R ( s ) ⩾ σ 2 and the cut-off resolvent R χ ( z ) has an analytic continuation for ℑ ( z ) < − i σ 2 , | R ( z ) | ⩾ C . To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Lefschetz Formulae for p-Adic Groups

Abstract In this paper, Lefschetz formulae for torus actions on p-adic groups are proven. These are similar to comparable formulae for real Lie groups. Applications lie in the realm of dynamical zeta functions.

Transfer operators for hard-rod-type models

We describe a new approach to the hard rod model and construct a Ruelle-Mayer-type transfer operator which satisfies a dynamical trace formula. This yields a meromorphic continuation of the dynamical zeta function in two variables among them the inverse temperature.

Patterson–Sullivan Distributions and Quantum Ergodicity

We relate two types of phase space distributions associated to eigenfunctions $\phi_{ir_j}$ of the Laplacian on a compact hyperbolic surface $X_{\Gamma}$: (1) Wigner distributions $\int_{S^*\X} a dW_{ir_j}=< Op(a)\phi_{ir_j}, \phi_{ir_j}>_{L^2(\X)}$, which arise in quantum chaos. They are invariant under the wave group. (2) Patterson-Sullivan distributions $PS_{ir_j}$, which are the residues of the dynamical zeta-functions $\lcal(s; a): = \sum_\gamma \frac{e^{-sL_\gamma}}{1-e^{-L_\gamma}} \int_{\gamma_0} a$ (where the sum runs over closed geodesics) at the poles $s = {1/2} + ir_j$. They are invariant under the geodesic flow. We prove that these distributions (when suitably normalized) are asymptotically equal as $r_j \to \infty$. We also give exact relations between them. This correspondence gives a new relation between classical and quantum dynamics on a hyperbolic surface, and consequently a formulation of quantum ergodicity in terms of classical ergodic theory.

Zeta functions for elements of entropy rank-one actions

An algebraic $\mathbb{Z}^d$-action of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.

Meromorphic extensions of a class of zeta functions for two-dimensional billiards without eclipse

The main purpose of the present paper is to show that a class of dynamical zeta functions associated with the so-called two-dimensional open billiard without eclipse have meromorphic extensions to the half-plane consisting of all complex numbers whose real parts are greater than a certain negative number. As an application, we verify that the zeta function for the length spectrum of the corresponding billiard table has the same property.

Multifractal analysis of the spectral measure of the Thue–Morse sequence: a periodic orbit approach

The Fourier spectral density of the Thue?Morse sequence is reinterpreted as the invariant measure of a stochastic dynamical system. Based on this fact, its generalized (R?nyi) dimension and f(?) statistics are calculated with high precision by cycle expansions of spectral determinant and dynamical zeta function. ?q at integer values of q are also computed in an operator scheme and the asymptotic result in the large-q limit is derived.