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Exponential inequalities and functional central limit theorems for random fields

Jérôme Dedecker (2010)

ESAIM: Probability and Statistics

We establish new exponential inequalities for partial sums of random fields. Next, using classical chaining arguments, we give sufficient conditions for partial sum processes indexed by large classes of sets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, the condition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixing random fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients. ...

Exponential mixing for the Teichmüller flow

Artur Avila, Sébastien Gouëzel, Jean-Christophe Yoccoz (2006)

Publications Mathématiques de l'IHÉS

We study the dynamics of the Teichmüller flow in the moduli space of abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the S L ( 2 , ℝ ) action in the moduli space has a spectral gap.

Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications

Tim Austin (2010)

Fundamenta Mathematicae

We study a generalized notion of a homogeneous skew-product extension of a probability-preserving system in which the homogeneous space fibres are allowed to vary over the ergodic decomposition of the base. The construction of such extensions rests on a simple notion of 'direct integral' for a 'measurable family' of homogeneous spaces, which has a number of precedents in older literature. The main contribution of the present paper is the systematic development of a formalism for handling such extensions,...

Extreme Relations for Topological Flows

Brunon Kamiński, Artur Siemaszko, Jerzy Szymański (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We introduce the concept of an extreme relation for a topological flow as an analogue of the extreme measurable partition for a measure-preserving transformation considered by Rokhlin and Sinai, and we show that every topological flow has such a relation for any invariant measure. From this result, it follows, among other things, that any deterministic flow has zero topological entropy and any flow which is a K-system with respect to an invariant measure with full support is a topological K-flow....

Faithful zero-dimensional principal extensions

Tomasz Downarowicz, Dawid Huczek (2012)

Studia Mathematica

We prove that every topological dynamical system (X,T) has a faithful zero-dimensional principal extension, i.e. a zero-dimensional extension (Y,S) such that for every S-invariant measure ν on Y the conditional entropy h(ν | X) is zero, and, in addition, every invariant measure on X has exactly one preimage on Y. This is a strengthening of the authors' result in Acta Appl. Math. [to appear] (where the extension was principal, but not necessarily faithful).

Fiber entropy and conditional variational principles in compact non-metrizable spaces

Tomasz Downarowicz, Jacek Serafin (2002)

Fundamenta Mathematicae

We consider a pair of topological dynamical systems on compact Hausdorff (not necessarily metrizable) spaces, one being a factor of the other. Measure-theoretic and topological notions of fiber entropy and conditional entropy are defined and studied. Abramov and Rokhlin's definition of fiber entropy is extended, using disintegration. We prove three variational principles of conditional nature, partly generalizing some results known before in metric spaces: (1) the topological conditional entropy...

Fibration of the phase space for the Korteweg-de Vries equation

Thomas Kappeler (1991)

Annales de l'institut Fourier

In this article we prove that the fibration of L 2 ( S 1 ) by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to N -gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.

Finitarily Bernoulli factors are dense

Stephen Shea (2013)

Fundamenta Mathematicae

It is not known if every finitary factor of a Bernoulli scheme is finitarily isomorphic to a Bernoulli scheme (is finitarily Bernoulli). In this paper, for any Bernoulli scheme X, we define a metric on the finitary factor maps from X. We show that for any finitary map f: X → Y, there exists a sequence of finitary maps fₙ: X → Y(n) that converges to f, where each Y(n) is finitarily Bernoulli. Thus, the maps to finitarily Bernoulli factors are dense. Let (X(n)) be a sequence of Bernoulli schemes such...

Finitary orbit equivalence and measured Bratteli diagrams

T. Hamachi, M. S. Keane, M. K. Roychowdhury (2008)

Colloquium Mathematicae

We prove a strengthened version of Dye's theorem on orbit equivalence, showing that if the transformation structures are represented as finite coordinate change equivalence relations of ergodic measured Bratteli diagrams, then there is a finitary orbit equivalence between these diagrams.

Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

Alexander I. Bufetov (2014)

Annales de l’institut Fourier

The main result of this note, Theorem 1.3, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant and ergodic under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of “corners" of finite size, must be finite. A similar result, Theorem 1.1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results imply that the infinite Hua-Pickrell measures...

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