Maximal ergodic theorems and applications to Riemannian geometry

We prove new ergodic theorems in the context of infinite ergodic theory, and give some applications to Riemannian and Kähler manifolds without conjugate points. One of the consequences of these ideas is that a complete manifold without conjugate points has nonpositive integral of the infimum of Ricci curvatures, whenever this integral makes sense. We also show that a complete Kähler manifold with nonnegative holomorphic curvature is flat if it has no conjugate points.     

The Mean-value of Meromorphic Functions with Respect to a Generalized Boolean Transformation

A formula for the mean-value distribution of certain meromorphic functions on a vertical line s = σ +iR under a generalized Boolean transformation, called rational Boolean transformation from R into itself, is derived using Birkhoff 's ergodic theorem. This formula is represented as a computable integral. Using the Cauchy's integral theorem, values of this integral corresponding to various possible cases are explicitly computed.     

Graph-theoretic and algebraic characterizations of some Markov processes

An algebraic decidable condition for a stationary Markov chain to consist of a single ergodic set, and a graph-theoretic decidable condition for a stationary Markov chain to consist of a single ergodic noncyclic set are formulated. In the third part of the paper a graph-theoretic condition for a nonstationary Markov chain to have the weakly-ergodic property is given. The paper is based on part of the author’s work towards the D. Sc. degree.     

广生灭过程的遍历性及平稳分布

文献[1]研究了广生灭过程的向上积分型随机泛函，得到了广生灭过程的若干数字特征以及常返的充要条件，该文讨论广生灭过程向下积分型随机泛函，给出了广生灭过程遍历的充要条件以及平均返回时间的计算公式，并在遍历的条件下求出了广生灭过程的平稳分布．     

A note on limit distributions of integral functions of on an ergodic Markov process

A theorem on an asymptotic behavior of integral functional of a nonperiodic ergodic Markov process is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1267–1269, September, 1991.     

Almost sure stability of linear stochastic differential equations with jumps

 Under the nondegenerate condition as in the diffusion case, see [14, 21, 6], the linear stochastic jump-diffusion process projected on the unit sphere is a strong Feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump-diffusions and the corresponding diffusions due to [22]. The unique deterministic Lyapunov exponent can be represented by the Furstenberg-Khas'minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign. The critical case of zero Lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated. Several examples are treated for illustration. Received: 22 June 2000 / Revised version: 20 November 2001 / Published online: 13 May 2002     

Quantum Ergodic Restriction Theorems. I: Interior Hypersurfaces in Domains with Ergodic Billiards

Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface H so that restrictions \({\phi_j |_H}\) to H of Δ-eigenfunctions of Riemannian manifolds (M, g) with ergodic geodesic flow are quantum ergodic on H. We prove two kinds of results: First (i) for any smooth hypersurface H in a piecewise-analytic Euclidean domain, the Cauchy data \({(\phi_j|H,\partial_{\nu}^H \phi_j|H)}\) is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly, (ii) we give conditions on H so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equality of left and right Poincaré maps for H. The proof involves two further novel results: (iii) a local Weyl law for boundary traces of eigenfunctions, and (iv) an ‘almost-orthogonality’ result for Fourier integral operators whose canonical relations almost nowhere commute with the geodesic flow.     

The Average Density of Self-Conformal Measures

The paper calculates the average density of the normalized Hausdorffmeasure on the fractal set generated by a conformal iteratedfunction system. It equals almost everywhere a positive constantgiven by a truncated generalized s-energy integral, where sis the corresponding Hausdorff dimension. As a main tool a conditionalGibbs measure is determined. The appendix proves an appropriateextension of Birkhoff's ergodic theorem which is also of independentinterest.     

On strictly ergodic models which are not almost topologically conjugate

Answering a question raised by Glasner and Rudolph (1984) we construct uncountably many strictly ergodic topological systems which are metrically isomorphic to a given ergodic system (X, ℬ,μ, T) but not almost topologically conjugate to it. This paper is part of the second author’s Ph.D. thesis, written under the supervision of Professor A. Bellow of the Department of Mathematics, Northwestern University. The author is grateful for her encouragement and advice. We acknowledge B. Weiss for helpful comments.     

Integral Self-affiine Tiles of Bandt＇s Model

Integral self-affine tiling of Bandt＇s model is a generalization of the integral self-affine tiling. Using ergodic theory, we show that the Lebesgue measure of the tile is a rational number where the denominator equals to the order of the associate symmetry group. We apply the result to the study of the Levy Dragon.     

Local-global principles for representations of quadratic forms

We prove a local-global principle for the problem of representations of quadratic forms by quadratic forms over ℤ, in codimension ≥5. The proof uses the ergodic theory of p-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.     

Ergodic properties of operators on integral points of ellipsoids

We prove that the operators on integral points of ellipsoids corresponding to the action of the group of classes of binary forms (see [7]) have ergodic properties, i.e., the sequence of such operators asymtpotically uniformly mixes the integral points over the set of translated lattices from one spinor genus and over the surface of the corresponding ellipsoids.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 159–175, 1986.     

Application of the discrete ergodic method to the analytical arithmetic of indefinite ternary quadratic forms

One obtains generalizations and refinements of ergodic and mixing theorems [Yu. V. Linnik, Ergodic Properties of Algebraic Fields, Springer-Verlag, New York (1968)] for the distribution of integral points on a two-sheeted hyperboloid. One gives a survey of the applications of the discrete ergodic method to indefinite ternary quadratic forms; in particular, one describes the results of the papers of A. M. Istamov, E. V. Podsypanin, A. V. Malyshev, and U. M. Pachev, published in this issue.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR, Vol. 93, pp. 5–24, 1980.     

Strong approximations and sequential change-point analysis for diffusion processes

In this paper ergodic diffusion processes depending on a parameter in the drift are considered under the assumption that the processes can be observed continuously. Strong approximations by Wiener processes for a stochastic integral and for the estimator process constructed by the one-step procedure of Le Cam are obtained. Applying these approximations, a CUSUM-type procedure is developed for the sequential testing of changes in the parameter.     

Ergodic theory and appication to nonconvex homogenization

We establish some ergodic theorems with the view to obtaining a convergence result of sequences of random Radon measures. We also give an application in stochastic homogenization of nonconvex integral functionals.     

L 1 ergodic behavior of non-negative kernels

Some results that have been obtained in the study of strongly and weakly ergodic behavior of non-homogeneous stochastic kernels are generalized to the case of non-negative kernels. The first generalization simply involves extending the definitions of weakly and strongly ergiodic behavior to the case of non-negative kernels and using the ergodic coefficient which was first defined for stochastic kernels by Dobrushin and extended to non-negative kernels by Blum and Reichaw. It happens that this straightforward extension excludes many cases of non-negative kernels which do exhibit a types of ergodic behavior. In order to study these cases a definition ofL ₁ weakly and strongly ergodic behavior is given in which normalizing by constants is allowed. Sufficient conditions for these types of ergodic behavior are given.     

A note on a variation of Doeblin's condition for uniform ergodicity of Markov chains

Summary We introduce a simple variation of Doeblin's condition, Condition D*, that assures the uniform ergodicity of a Markov chain. It is also shown that for non-homogeneous chains our conditions are equivalent to Dobrushin's weak ergodic coefficient.     

Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model

We discuss the inviscid limits for the randomly forced 2D Navier-Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier-Stokes equation, written in terms of the ergodic measures.     

Uniform distribution of integral points on 3-dimensional spheres via modular forms

The problem of the asymptotic distribution of integral points on a sequence of expanding spheres x² + y² + z² = m_i [first penetrated by Linnik with his ergodic method] is shown to be attackable by analytic methods, especially via the theory of modular forms. Success depends on a new estimate for the Fourier coeff. of cusp forms of half integral dimension on the theta group, which we obtain after improving upon previously known estimates of the associated Kloostermann sums.     

Faces of simplexes of invariant measures

This paper is a direct continuation of [D2]. The main result proved here, combined with Theorem 1 of [D2] widens the variety of known possibilities of what the simplex of invariant measures in a minimal topological dynamical system can be. We show that it can be equivalent, modulo affine homeomorphism of Choquet simplexes and modulo isomorphisms of the ergodic measures playing the role of the extreme points, to any face of the simplex of invariant measures of any zero-dimensional topological dynamical system, as long as this face contains no periodic measures. In particular, this implies that any formal simplex spanned by any choice of countably many nonperiodic ergodic measures can be realized in a minimal system. The first draft of this paper was written during the author’s the author’s visit at Université Paris 6. Research supported from resources for science in years 2005–2008 as research project (grant MENII 1 P03A 021 29, Poland)