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A characterization of partition polynomials and good Bernoulli trial measures in many symbols

Andrew Yingst (2014)

Colloquium Mathematicae

Consider an experiment with d+1 possible outcomes, d of which occur with probabilities x ₁ , . . . , x d . If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in x ₁ , . . . , x d . We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

A cut salad of cocycles

Jon Aaronson, Mariusz Lemańczyk, Dalibor Volný (1998)

Fundamenta Mathematicae

We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.

A note on a generalized cohomology equation

K. Krzyżewski (2000)

Colloquium Mathematicae

We give a necessary and sufficient condition for the solvability of a generalized cohomology equation, for an ergodic endomorphism of a probability measure space, in the space of measurable complex functions. This generalizes a result obtained in [7].

An anti-classification theorem for ergodic measure preserving transformations

Matthew Foreman, Benjamin Weiss (2004)

Journal of the European Mathematical Society

Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT’s on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos–von Neumann theorem classifying ergodic MPT’s with pure point spectrum. We treat both the classical case of...

An Application of Skew Product Maps to Markov Chains

Zbigniew S. Kowalski (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

By using the skew product definition of a Markov chain we obtain the following results: (a) Every k-step Markov chain is a quasi-Markovian process. (b) Every piecewise linear map with a Markovian partition defines a Markov chain for every absolutely continuous invariant measure. (c) Satisfying the Chapman-Kolmogorov equation is not sufficient for a process to be quasi-Markovian.

Asymptotic Vassiliev invariants for vector fields

Sebastian Baader, Julien Marché (2012)

Bulletin de la Société Mathématique de France

We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of ℝ 3 . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.

Cardinality of Rauzy classes

Vincent Delecroix (2013)

Annales de l’institut Fourier

Rauzy classes form a partition of the set of irreducible permutations. They were introduced as part of a renormalization algorithm for interval exchange transformations. We prove an explicit formula for the cardinality of each Rauzy class. Our proof uses a geometric interpretation of permutations and Rauzy classes in terms of translation surfaces and moduli spaces.

Central limit theorems for non-invertible measure preserving maps

Michael C. Mackey, Marta Tyran-Kamińska (2008)

Colloquium Mathematicae

Using the Perron-Frobenius operator we establish a new functional central limit theorem for non-invertible measure preserving maps that are not necessarily ergodic. We apply the result to asymptotically periodic transformations and give a specific example using the tent map.

Conjugacies between ergodic transformations and their inverses

Geoffrey Goodson (2000)

Colloquium Mathematicae

We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation S T = T - 1 S . In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of S 2 . In particular, S 2 has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace f ∈ L 2 ( X , ℱ , μ ) : f ( T 2 x ) = f ( x ) . For S and T ergodic satisfying this equation further constraints arise,...

Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Jean-Pierre Conze, Albert Raugi (2003)

ESAIM: Probability and Statistics

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet” condition and apply it to a class of transition operators. This gives the convergence of the series ∑ k ≥ 0 k r P k f , r ∈ ℕ , under some regularity assumptions and implies the central limit theorem with a rate in n - 1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

Convergence of iterates of a transfer operator, application to dynamical systems and to Markov chains

Jean-Pierre Conze, Albert Raugi (2010)

ESAIM: Probability and Statistics

We present a spectral theory for a class of operators satisfying a weak “Doeblin–Fortet" condition and apply it to a class of transition operators. This gives the convergence of the series ∑k≥0krPkƒ, r ∈ ℕ , under some regularity assumptions and implies the central limit theorem with a rate in n - 1 2 for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.

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