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Weakly mixing but not mixing quasi-Markovian processes

Zbigniew Kowalski (2000)

Studia Mathematica

Let (f,α) be the process given by an endomorphism f and by a finite partition α = A i i = 1 s of a Lebesgue space. Let E(f,α) be the class of densities of absolutely continuous invariant measures for skew products with the base (f,α). We say that (f,α) is quasi-Markovian if E ( f , α ) ⊂ g : ⋁ B i i = 1 s s u p p g = ⋃ i = 1 s A i × B i . We show that there exists a quasi-Markovian process which is weakly mixing but not mixing. As a by-product we deduce that the set of all coboundaries which are measurable with respect to the ’chequer-wise’ partition for σ × S, where σ is...

Weakly mixing rank-one transformations conjugate to their squares

Alexandre I. Danilenko (2008)

Studia Mathematica

Utilizing the cut-and-stack techniques we construct explicitly a weakly mixing rigid rank-one transformation T which is conjugate to T². Moreover, it is proved that for each odd q, there is such a T commuting with a transformation of order q. For any n, we show the existence of a weakly mixing T conjugate to T² and whose rank is finite and greater than n.

Weighted Lp spaces and pointwise ergodic theorems.

Ryotaro Sato (1995)

Publicacions Matemàtiques

In this paper we give an operator theoretic version of a recent result of F. J. Martín-Reyes and A. de la Torre concerning the problem of finding necessary and sufficient conditions for a nonsingular point transformation to satisfy the Pointwise Ergodic Theorem in Lp. We consider a positive conservative contraction T on L1 of a σ-finite measure space (X, F, μ), a fixed function e in L1 with e > 0 on X, and two positive measurable functions V and W on X. We then characterize the pairs (V,W)...

Which Bernoulli measures are good measures?

Ethan Akin, Randall Dougherty, R. Daniel Mauldin, Andrew Yingst (2008)

Colloquium Mathematicae

For measures on a Cantor space, the demand that the measure be "good" is a useful homogeneity condition. We examine the question of when a Bernoulli measure on the sequence space for an alphabet of size n is good. Complete answers are given for the n = 2 cases and the rational cases. Partial results are obtained for the general cases.

Zero Krengel entropy does not kill Poisson entropy

Élise Janvresse, Thierry de la Rue (2012)

Annales de l'I.H.P. Probabilités et statistiques

We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann–Kakutani odometer), but whose associated Poisson suspension has positive entropy.

Zeros of random functions in Bergman spaces

Joel H. Shapiro (1979)

Annales de l'institut Fourier

Suppose μ is a finite positive rotation invariant Borel measure on the open unit disc Δ , and that the unit circle lies in the closed support of μ . For 0 < p < ∞ the Bergman space A μ p is the collection of functions in L p ( μ ) holomorphic on Δ . We show that whenever a Gaussian power series f ( z ) = Σ ζ n a n z n almost surely lies in A μ p but not in ⋃ q > p A μ p , then almost surely: a) the zero set Z ( f ) of f is not contained in any A μ q zero set ( q > p , and b) Z ( f + 1 ) ∪ Z ( f - 1 ) is not contained in any A μ q zero set.

α-Equivalence

Kyewon Koh Park (1998)

Studia Mathematica

We define the α - relations between discrete systems and between continuous systems. We show that it is an equivalence relation. α- Equivalence vs. even α-equivalence is analogous to Kakutani equivalence vs. even Kakutani equivalence.

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