On the ergodic theory of non-integrable functions and infinite measure spaces

We prove a Chow-Robbins type result for an ergodic, non-negative SSSP, and a similar result for transformations preserving infinite measure, which implies that for these transformations, no “absolute” version of Hopf's theorem can hold.     

Generators

In this note we indicate the importance of the notion of “generator” for the classification problem in ergodic theory, and we give a simple new construction of a finite generator for an ergodic transformation of finite entropy This revised version was published online in June 2006 with corrections to the Cover Date.     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.     

Orthogonal and bounded orthogonal spectral representations

The concept of an orthogonal spectral representation (OTSR) of a Hilbert spaceH relative to a spectral measureE(.) is introduced and it is shown that every Hilbert space admits an OTSR relative to a given spectral measure. Apart from the various results obtained about OTSRs, the principal result of Allan Brown (1974) is deduced as an easy consequence of this study. A new complete system of unitary invariants called the “equivalence of OTSRs”, is given for spectral measures. Two special types of OTSRs called “BOTSR” and “COBOTSR” are introduced and characterized respectively in terms of the “GCGS-property” and “CGS-property” of the associated spectral measure. Various complete systems of unitary invariants are given for spectral measures with the GCGS-property. Finally, the Wecken-Plesner-Rohlin theorem on hermitian operators with simple spectra is generalized to arbitrary spectral measures.     

Noncommutative Jordan superalgebras of degree <Emphasis Type="Italic">n</Emphasis> > 2

We prove a coordinatization theorem for noncommutative Jordan superalgebras of degree n > 2, describing such algebras. It is shown that the symmetrized Jordan superalgebra for a simple finite-dimensional noncommutative Jordan superalgebra of characteristic 0 and degree n > 1 is simple. Modulo a “nodal” case, we classify central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0.     

Spectral Deviations for the Damped Wave Equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the “entropy” that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.     

Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Transplantation harmonique,transplantation par modules,et théorèmes isopérimétriques

Summary The “harmonic transplantation” allows to extend some isoperimetric theorems, so far proved by conformal mapping, to higher connectivity and to higher dimensions; for the first eigenvalue λ₁ of a membrane, it again can give only upper bounds.—The “transplantation by moduli” is much more flexible; for example, it leads to a simple one-dimensional interpretation of the Rayleigh-Faber-Krahn theorem.      

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

Random ergodic theorems and real cocycles

We study mean convergence of ergodic averages associated to a measure-preserving transformation or flow τ along the random sequence of times κ _n (ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT. We prove that the sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations. When no assumption is made onF, the random sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k _n(ω)) is not almost surely universally good for the class of weakly mixing transformations.     

On Sobolev solutions of poisson equations in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> with a parameter

Smoothness with respect to a parameter is established under mild assumptions on the regularity of coefficients for Sobolev solutions of the Poisson equations in the whole ℝ ^d in the “ergodic case.” An assertion of this kind serves as one of the key tools in diffusion approximation and some other limit theorems. Bibliography: 12 titles.     

On the cohomology of ergodic group actions

We consider three problems concerning cocycles of ergodic group actions: behavior of cohomology under the restriction of an ergodic semi-simple Lie group action to a lattice subgroup; “compactness” of bounded cocyles; and the relation of relative almost periodicity to relative discrete spectrum for extensions of ergodic actions.     

The majority decision function for trees with 3 leaves

Kenneth May in 1952 proved a classical theorem characterizing simple majority rule for two alternatives. The present paper generalizes May’s theorem to the case of three alternatives, but where the voters’ preference relations are required to be trees with alternatives at the leaves. This paper with title “May’s Theorem for Trees” was presented at the joint DIMACS-LAMSADE workshop on Computer Science and Decision Theory, Université Paris-Dauphine, Paris 27–29 October 2004. It appeared in the proceedings Annales du LAMSADE 3 (2004), 259–266.     

Reducing prime graphs and recognizing circle graphs

We prove a reduction theorem for prime (simple) graphs in Cunningham’s sense. Roughly speaking this theorem says that every prime (simple) graph of ordern>5 “contains” a smaller prime graph of ordern−1. As an application we give a polynomial algorithm for recognizing circle graphs.     

Recurrence and ergodicity of interacting particle systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Received: 22 January 1999 / Revised version: 24 May 1999     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups

We show that if (S(t)) _t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically isometric” almost-orbit of (S(t)) _t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions.     

On almost 1-1 extensions

We show that a broad class of extensions of measure preserving systems in the context of ergodic theory can be realized by topological models for which the extension is “almost one-one”.     

A new proof of the Amitsur-Levitski identity

We give a simple proof of the Amitsur-Levitzki identity by analysing the powers of matrices with “differential 1-forms” as entries. Using the fact that 2-forms are central the identity is seen to follow from the Cayley-Hamilton theorem.