### A class of generalized Ornstein transformations with the weak mixing property

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The main result of this paper is a quantified version of Ingham's Tauberian theorem for bounded vector-valued sequences rather than functions. It gives an estimate on the rate of decay of such a sequence in terms of the behaviour of a certain boundary function, with the quality of the estimate depending on the degree of smoothness this boundary function is assumed to possess. The result is then used to give a new proof of the quantified Katznelson-Tzafriri theorem recently obtained by the author...

For a sequence of dependent random variables ${\left({X}_{k}\right)}_{k\in \mathbb{N}}$ we consider a large class of summability methods defined by R. Jajte in [jaj] as follows: For a pair of real-valued nonnegative functions g,h: ℝ⁺ → ℝ⁺ we define a sequence of “weighted averages” $1/g\left(n\right){\sum}_{k=1}^{n}\left({X}_{k}\right)/h\left(k\right)$, where g and h satisfy some mild conditions. We investigate the almost sure behavior of such transformations. We also take a close look at the connection between the method of summation (that is the pair of functions (g,h)) and the coefficients that measure...

We give an example of a dynamical system which is mixing relative to one of its factors, but for which relative mixing of order three does not hold.

We study a nonconventional ergodic average for asymptotically abelian weakly mixing C*-dynamical systems, related to a second iteration of Khinchin's recurrence theorem obtained by Bergelson in the measure-theoretic case. A noncommutative recurrence theorem for such systems is obtained as a corollary.

We combine some results from the literature to give examples of completely mixing interval maps without limit measure.

${M}_{1}$ è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale $K$ che è, in un certo senso, una sorta di frontiera di $K$. Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su $K$ un analogo del funzionale $u\mapsto \int {\left|\text{grad}u\right|}^{2}$ e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a ${M}_{1}$ ), e sulla convergenza dell'iterata di ${M}_{1}$ rinormalizzata. Questi risultati sono collegati con l'unicità...

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 ${\sum}_{k\ge 0}{k}^{r}{P}^{k}f$, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in ${n}^{-\frac{1}{2}}$ 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, $r\in \mathbb{N}$, under some regularity assumptions and implies the central limit theorem with a rate in ${n}^{-\frac{1}{2}}$ for the corresponding Markov chain. An application to a non uniformly hyperbolic transformation on the interval is also given.