Equilibrium states for the geodesic flow

We give in this paper new results of large deviation type for the geodesic flow on a closed Riemannian manifold, which describe the proportion of geodesic arcs supporting measures close to the equilibrium states. We introduce zeta functions in terms of geodesic arcs and show that they define holomorphic functions on a half plane given by the topological pressure.     

An approach to the zeta function of a finite factorgraph via the Markov topological chains

An approach to the zeta and the L-function of a finite connected graph via the theory of Markov topological chains (symbolic dynamics) is presented. The case of the L-function of a finite connected graph with labeling is considered. Bibliography: 12 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 226–245. Translated by A. M. Nikitin.     

Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle–Perron–Frobenius operator acting on the space of Hölder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.     

Extensions of Zeta Functions – Examples and a Study of the Double Sine Functions

We discuss zeta extensions in the sense of Kurokawa and Wakayama, Proc. Japan Acad. 2002, for constructing new zeta functions from a given zeta function. This notion appeared when we introduced higher zeta functions such as higher Riemann zeta functions in Kurokawa et al., Kyushu Univ. Preprint, 2003, and a higher Selberg zeta functions in Kurokawa and Wakayama, Comm. Math. Phys., 2004. In this article, we first recall some explicit examples of such zeta extensions and give a conjecture about functional equations satisfied by higher zeta functions. We devote the second part to making a detailed study of the double sine functions which are treated in a framework of the zeta extensions.Mathematics Subject Classification (2000) 11M36.Partially supported by Grant-in-Aid for Scientific Research (B) No. 15340012, and by Grant-in-Aid for Exploratory Research No. 13874004. This is based on the talk at The 2002 Twente Conference on Lie Groups 16–18 Dec. University of Twente, Enschede, The Netherlands.     

The Denef-Loeser Zeta Function is not a Topological Invariant

An example is given which shows that the Denef–Loeserzeta function (usually called the topological zeta function)associated to a germ of a complex hypersurface singularity isnot a topological invariant of the singularity. The idea isthe following. Consider two germs of plane curves singularitieswith the same integral Seifert form but with different topologicaltype and which have different topological zeta functions. Makea double suspension of these singularities (consider them ina 4-dimensional complex space). A theorem of M. Kervaire andJ. Levine states that the topological type of these new hypersurfacesingularities is characterized by their integral Seifert form.Moreover the Seifert form of a suspension is equal (up to sign)to the original Seifert form. Hence these new singularitieshave the same topological type. By means of a double suspensionformula the Denef–Loeser zeta functions are computed forthe two 3-dimensional singularities and it is verified thatthey are not equal.     

Existence,uniqueness and stability of equilibrium states for non-uniformly expanding maps

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact metric spaces and Hölder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.     

Thermodynamic formalism for null recurrent potentials

We extend Ruelle’s Perron-Frobenius theorem to the case of Hölder continuous functions on a topologically mixing topological Markov shift with a countable number of states. LetP(?) denote the Gurevic pressure of ? and letL _? be the corresponding Ruelle operator. We present a necessary and sufficient condition for the existence of a conservative measure ν and a continuous functionh such thatL _? ^* ν=e ^P(?)ν,L _? h=e ^P(?) h and characterize the case when ∝hdν<∞. In the case whendm=hdν is infinite, we discuss the asymptotic behaviour ofL _? ^k , and show how to interpretdm as an equilibrium measure. We show how the above properties reflect in the behaviour of a suitable dynamical zeta function. These resutls extend the results of [18] where the case ∝hdν<∞ was studied.     

Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms

The theory of zeta functions associated with prehomogeneous vector spaces (p.v. for short) provides us a unified approach to functional equations of a large class of zeta functions. However the general theory does not include zeta functions related to automorphic forms such as the HeckeL-functions and the standardL-functions of automorphic forms on GL(n), even though they can naturally be considered to be associated with p.v.’s. Our aim is to generalize the theory to zeta functions whose coefficients involve periods of automorphic forms, which include the zeta functions mentioned above. In this paper, we generalize the theory to p.v.’s with symmetric structure ofK _ε-type and prove the functional equation of zeta functions attached to automorphic forms with generic infinitesimal character. In another paper, we have studied the case where automorphic forms are given by matrix coefficients of irreducible unitary representations of compact groups. Dedicated to the memory of Professor K G Ramanathan     

On sofic systems II

The results ofOn sofic systems I on topological Markov chains extending sofic systems are completed. To homomorphisms of sofic systems are canonically associated homomorphisms of Markov extensions. Also considered is a class of finitary codes for sofic systems.     

Motivic zeta functions of infinite-dimensional Lie algebras

The paper shows how to associate a motivic zeta function with a large class of infinite dimensional Lie algebras. These include loop algebras, affine Kac-Moody algebras, the Virasoro algebra and Lie algebras of Cartan type. The concept of a motivic zeta functions provides a good language to talk about the uniformity in p of local p-adic zeta functions of finite dimensional Lie algebras. The theory of motivic integration is employed to prove the rationality of motivic zeta functions associated to certain classes of infinite dimensional Lie algebras.     

Stochastic model of phase transition and metastability

The evolution of a system with phase transition is simulated by a Markov process whose transition probabilities depend on a parameter. The change of the stationary distribution of the Markov process with a change of this parameter is interpreted as a phase transition of the system from one thermodynamic equilibrium state to another. Calculations and computer experiments are performed for condensation of a vapor. The sample paths of the corresponding Markov process have parts where the radius of condensed drops is approximately constant. These parts are interpreted as metastable states. Two metastable states occur, initial (gaseous steam) and intermediate (fog). The probability distributions of the drop radii in the metastable states are estimated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 94–106, April, 2000.     

Completely integrable nonlinear dynamical systems of the Langmuir chains type

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Padé approximations, and on the Sturm-Liouville method for finite difference equations is used. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 588–602, October, 1997. Translated by A. M. Chebotarev     

A topological structure on a set of continous functions with various domains

We construct a topological space of continuous functions which generalizes the previously studied space of functions defined on closed intervals. For the new space, metrizability properties are studied. The results can be applied in the topological theory of ordinary differential equations. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 76–88, July, 1999.     

Nonindependent splittings and Gibbs states

We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 598–605, October, 1998.This research was partially supported by the International Science Foundation under grant No. 96-0698     

A χ2 goodness-of-fit test for Markov renewal processesgoodness-of-fit test for Markov renewal processes

A Markov Renewal Process (M.R.P.) is a process similar to a Markov chain, except that the time required to move from one state to another is not fixed, but is a random variable whose distribution may depend on the two states between which the transition is made. For an M.R.P. ofm (<∞) states we derive a goodness-of-fit test for a hypothetical matrix of transition probabilities. This test is similar to the test Bartlett has derived for Markov chains. We calculate the first two moments of the test statistic and modify it to fit the moments of a standard χ². Finally, we illustrate the above procedure numeerically for a particular case of a two-state M.R.P. Dwight B. Brock is mathematical statistican, Office of Statistical Methods, National Center for Health Statistics, Rockville, Maryland. A. M. Kshisagar is Associate Professor, Department of Statistics, Southern Methodist University. This research was partially supported by Office of Naval Research Contract No. N000 14-68-A-0515, and by NIH Training Grant GM-951, both with Southern Methodist University. This article is partially based on Dwight B. Brock's Ph.D. dissertation accepted by Southern Methodist University.     

The Factorization of Queueing Equations and Their Interpretation

There are many queueing systems, including the M ^x /M ^y /c queue, the GI ^x /M/c queue and the M/D/c queue, in which the distribution of the queue length at certain epochs is determined by a Markov chain with the following structure. Except for a number of boundary states, all columns of the transition matrix are identical except for a shift which assures that there is always the same element occupying the main diagonal. This paper describes how one can find the equilibrium distribution for such Markov chains. Typically, this problem is solved by factorizing of a certain expression characterizing the repeated columns. In this paper, we show that this factorization has a probabilistic significance and we use this result to develop new approaches for finding the equilibrium distribution in question.     

Poles of the topological zeta function associated to an ideal in dimension two

To an ideal in one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an embedded resolution of the curve. In this paper we will study two questions about the poles of this zeta function. First, we will give a criterion to determine whether or not a candidate pole is a pole. It turns out that we can know this immediately by looking at the intersection diagram of the principalization, together with the numerical data of the exceptional curves. Afterwards we will completely describe the set of rational numbers that can occur as poles of a topological zeta function associated to an ideal in dimension two. The same results are valid for related zeta functions, as for instance the motivic zeta function. The research was partially supported by the Fund of Scientific Research—Flanders (G.0318.06).     

KMS states and complex multiplication

We construct a quantum statistical mechanical system which generalizes the Bost–Connes system to imaginary quadratic fields K of arbitrary class number and fully incorporates the explicit class field theory for such fields. This system admits the Dedekind zeta function as partition function and the idèle class group as group of symmetries. The extremal KMS states at zero temperature intertwine this symmetry with the Galois action on the values of the states on the arithmetic subalgebra. The geometric notion underlying the construction is that of commensurability of K-lattices.     

On the Poles of Maximal Order of the Topological Zeta Function

The global and local topological zeta functions are singularityinvariants associated to a polynomial f and its germ at 0, respectively.By definition, these zeta functions are rational functions inone variable, and their poles are negative rational numbers.In this paper we study their poles of maximal possible order.When f is non-degenerate with respect to its Newton polyhedron,we prove that its local topological zeta function has at mostone such pole, in which case it is also the largest pole; wegive a similar result concerning the global zeta function. Moreover,for any f we show that poles of maximal possible order are alwaysof the form –1/N with N a positive integer. 1991 MathematicsSubject Classification 14B05, 14E15, 32S50.