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Consider an experiment with d+1 possible outcomes, d of which occur with probabilities . 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 . 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.
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.
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].
We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these.
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...
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.
We analyse the asymptotical growth of Vassiliev invariants on non-periodic flow lines of ergodic vector fields on domains of . More precisely, we show that the asymptotics of Vassiliev invariants is completely determined by the helicity of the vector field.
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.
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.
We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation . 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 . In particular, has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace . For S and T ergodic satisfying this equation further constraints arise,...
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 , , under some regularity assumptions and implies the central limit theorem with a rate in for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.
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,
,
under some regularity assumptions and implies the central limit theorem
with a rate in for the corresponding Markov chain.
An application to a non uniformly hyperbolic transformation on the
interval is also given.
We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently by B. Host.
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