Ergodic Flows Research Articles

Metric with ergodic geodesic flow is completely determined by unparameterized geodesics

Let g g be a Riemannian metric with ergodic geodesic flow. Then if some metric g ¯ \bar g has the same geodesics (regarded as unparameterized curves) with g g , then the metrics are homothetic. If two metrics on a closed surface of genus greater than one have the same geodesics, then they are homothetic.

Unique ergodicity on compact homogeneous spaces

Extending results of a number of authors, we prove that if U U is the unipotent radical of an R \mathbb {R} -split solvable epimorphic subgroup of a real algebraic group G G which is generated by unipotents, then the action of U U on G / Γ G/\Gamma is uniquely ergodic for every cocompact lattice Γ \Gamma in G G . This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are the Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the ‘Cone Lemma’) about representations of epimorphic subgroups.

Billiards in infinite polygons

We introduce billiards in polygons with an infinite number of sides. We show that for almost every point the billiard flow in an infinite polygon is defined for all times and the Poincaré recurrence theorem holds. Our main results are that ergodic infinite polygons are typical among all infinite polygons and rational infinite polygons with ergodic directional flows are typical among all rational infinite polygons.

Convection–diffusion equation with space–time ergodic random flow

We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As a consequence, we obtain both upper and lower bounds on the effective diffusivity.

Rate of quantum ergodicity in Euclidean billiards

For a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum-mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine, and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing-ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a Gaussian distribution.

Considerations on diagnostics during operation with the dynamic ergodic divertor on TEXTOR-94

In this contribution, at first the required diagnostics are described according to the physics goals such as the study of particle removal, performance optimization, near field divertor option, island and ergodic structure and flows and possible confinement improvement due to the field rotation. In a second part, details of the different diagnostics are discussed.

Embedded Surfaces with Ergodic Geodesic Flows

Following ideas of Osserman, Ballmann and Katok, we construct smooth surfaces with ergodic, and indeed Bernoulli, geodesic flow that are isometrically embedded in R3. These surfaces can have arbitrary genus and can be made analytic.

Ergodicity of billiards in polygons

In the space of all polygons, a topologically massive subset consisting of polygons with ergodic billiard flows is explicitly described. The elements of this set have a specified order of approximation by rational polygons. As intermediate results, constructive versions of the ergodic theorem for the billiard in a rational polygon and for the geodesic flow on a surface with flat structure, and also a constructive quadratic estimate for the growth of the number of saddle connections (singular trajectories) in a flat structure, are proved.

Nilpotent flows of S1-invariant Hamiltonian systems on 4-dimensional symplectic manifolds

We investigate S1-invariant Hamiltonian systems on compact 4-dimensional symplectic manifolds with free symplectic action of a circle. We show that, in a rather general case, such systems generate ergodic flows of types (quasiperiodic and nilpotent) on their isoenergetic surfaces. We solve the problem of straightening of these flows.

Vibrational relaxation and energy localization in polyatomics: Effects of high-order resonances on flow rates and the quantum ergodicity transition

That high-order resonances may be involved in determining rates of vibrational relaxation in highly excited molecules has been recognized for some time. In this paper, high-order resonances are shown to play a central role in establishing the transition to ergodic flow, which we call the quantum ergodicity transition, at low energy in many-dimensional systems. The location of the transition to global vibrational flow in a system of coupled nonlinear oscillators is found to be generally sensitive to high-order resonances. On the other hand, except immediately above the transition, rates of flow once established depend primarily on the familiar low-order terms until the molecule reaches much higher excitation. Both the orders of resonant coupling terms in the vibrational Hamiltonian important in determining the transition to ergodicity, and the orders relevant for determining vibrational relaxation rates in large molecules are found to scale in a simple way with the strength of resonant coupling terms and the size of the molecule.

Toeplitz flows with pure point spectrum

We construct strictly ergodic 0-1 Toeplitz flows with pure point spectrum and irrational eigenvalues. It is also shown that the property of being regular is not a measure-theoretic invariant for strictly ergodic Toeplitz flows.

Mutual isomorphisms of translations of a homogeneous flow

Ergodic one-parameter flows (G/Γ,gℝ) induced by the left action of a subgroupgℝ ⊂G on homogeneous spaces of finite volume are considered. Letℳ ⊂ ℝ+ be the set of allt>0 such that the cascade (G/Γ,gtℤ) is metrically isomorphic to the cascade (G/Γ,gℤ). We prove that eitherℳ is at most countable or the subgroupgℝ is horocyclic andℳ=ℝ+. We prove that a metric isomorphism of ergodic quasi-unipotent cascades (or flows) is affine on almost all fibers of a certain natural bundle. The result generalizes Witte's theorem on the affinity of such isomorphisms of cascades with the mixing property; this is applied to the study of the structure of the setℳ ⊂ ℝ+. The proof is based on the fundamental Ratner theorem stating that the ergodic measures of unipotent cascades are algebraic.

Relatively perfect σ-algebras for flows

We show that for every ergodic flow, given any factor σ-algebra ℱ, there exists a σ-algebra which is relatively perfect with respect to ℱ. Using this result and Ornstein's isomorphism theorem for flows, we give a functorial definition of the entropy of fl

The Inverse Stable Range Functor

We give an inverse construction of the stable range for general flows which may or may not admit an invariant measure. The inverse map is then shown to be a right inverse functor of the stable range functor. The Krieger theorem asserts that in the type IIIo case, there is a one-toone correspondence between orbit equivalent classes of ergodic transformations, algebraic isomorphism classes of Krieger factors, and conjugate isomorphism classes of nontransitive ergodic flows. This correspondence is established via explicit maps between the objects, namely the stable range map, the crossed product, and the flow of weights. In this paper we are concerned with inverting the stable range map, i.e., given a flow on a standard measure space, we will explicitly construct a transformation with the flow as its stable range. Since orbit equivalent transformations give rise to conjugate flows under the stable range map, the choice of the transformation is not unique. However, by using some kind of skew product with the odometer, it is possible to choose a natural one which satisfies the requirement. This construction provides the vital missing arrow which points in the opposite direction of the other maps. For the case where the flow admits an invariant measure, such a transformation can be found in [4] or [5], but for a general flow the problem is more difficult. In [3] Hamachi showed that for a given ergodic flow, one can construct an ergodic action of Z x (Z + aZ) on a certain standard measure space whose stable range is isomorphic to the flow. The result of Connes, Feldman, and Weiss guarantees that the equivalence relation generated by this action is generated by a single transformation T, but it is far from clear how to construct such a transformation explicitly. Noting that the stable range map is in fact a functor from the category of transformations with orbit transporting isomorphisms to the category of flows with conjugations, we see that the construction below also extends to the construction of a right inverse functor to the stable range functor. This fact establishes that the mod homomorphism from the automorphism group of a measured equivalence relation to the automorphism group of its associated flow is split (this also follows from Hamachi's construction [3, p. 399]). Our method also yields a transparent proof of surjectivity of mod, a fact noted by both Hamachi and Golodets [1]. Received by the editors June 8, 1990 and, in revised form, November 7, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D99. ? 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

Ergodic geodesic flows on product manifolds with low dimensional factors.

Article Ergodic geodesic flows on product manifolds with low dimensional factors. was published on January 1, 1994 in the journal Journal für die reine und angewandte Mathematik (volume 1994, issue 450).

ON QUOTIENTS OF NONSINGULAR ACTIONS WHOSE SELF-JOININGS ARE GRAPHS

We study the notions of minimal self-joinings (MSJ) and graph self-joinings (GSJ) (analogous to simplicity in the finite measure preserving case) for nonsingular actions of locally compact groups. We show that for a nonsingular action of a group G with MSJ, every quotient comes from a closed subgroup of the center of G whose action is totally non-ergodic. Thus, totally ergodic nonsingular flows with MSJ are prime. We then show an analogous result to Veech's theorem, namely that for a nonsingular action of a group G with GSJ, every quotient comes from a locally compact subgroup of the centralizer whose action is totally non-ergodic.

On the rate of quantum ergodicity I: Upper bounds

One problem in quantum ergodicity is to estimate the rate of decay of the sums $$S_k (\lambda ;A) = \frac{1}{{N(\lambda )}}\sum\limits_{\sqrt {\lambda _j } \leqq \lambda } {\left| {(A\varphi _j ,\varphi _j ) - \bar \sigma _A } \right|^k } $$ on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ j ,ϕ j } are the spectral data of the Δ of(M, g), A is a 0-th order ψDO, $$\bar \sigma _A $$ is the (Liouville) average of its principal symbol and $$N(\lambda ) = \# \{ j:\sqrt {\lambda _j } \leqq \lambda \} $$ . ThatS k (λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatS k (λ;A)=O((logλ) −k/2 ) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).

On the rate of Quantum Ergodicity, II: Lower bounds

One problem in quantum ergodicity is to estimate the rate of decay of the sums $$S_k (\lambda ;A) = \frac{1}{{N(\lambda )}}\sum\limits_{\sqrt {\lambda _j } \leqq \lambda } {\left| {(A\varphi _j ,\varphi _j ) - \bar \sigma _A } \right|^k } $$ on a compact Riemannian manifold (M, g) with ergodic geodesic flow. Here, {λ j ,ϕ j } are the spectral data of the Δ of(M, g), A is a 0-th order ψDO,\(\bar \sigma _A \) is the (Liouville) average of its principal symbol and\(N(\lambda ) = \# \{ j:\sqrt {\lambda _j } \leqq \lambda \} \). ThatS k (λ;A)=o(1) is proved in [S, Z.1, CV.1]. Our purpose here is to show thatS k (λ;A)=O((logλ) −k/2 ) on a manifold of (possibly variable) negative curvature. The main new ingredient is the central limit theorem for geodesic flows on such spaces ([R, Si]).

Moving Weighted Averages

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

Ergodic theory, irreversibility and relativistic symmetry

Two fundamental aspects of nature are: (1) relativistic symmetry under frame transformations, and (2) time irreversibility of complex, dynamical systems. In this paper we first examine the relativistic transformation theory of some complex, dynamical systems encountered in ergodic theory; specifically, we prove that ergodic flows, mixing flows and K-flows are all relativistic invariants. Similarly, we examine the relativistic symmetry of the Brussels School's “microscopic theory of irreversibility”, and show that this theory of irreversibility is compatible with the basic tenets of special relativity.