Wyniki wyszukiwania

1

Some examples of cocycles with simple continuous singular spectrum

100%

Frączek K.

Studia Mathematica

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2001

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tom 146

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nr 1

1-13

EN

We study spectral properties of Anzai skew products $T_{φ}: 𝕋² → 𝕋²$ defined by $T_{φ}(z,ω) = (e^{2πiα}z,φ(z)ω)$, where α is irrational and φ: 𝕋 → 𝕋 is a measurable cocycle. Precisely, we deal with the case where φ is piecewise absolutely continuous such that the sum of all jumps of φ equals zero. It is shown that the simple continuous singular spectrum of $T_{φ}$ on the orthocomplement of the space of functions depending only on the first variable is a "typical" property in the above-mentioned class of cocycles, if α admits a sufficiently fast approximation.

2

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

63%

Frączek K., Wysokińska M.

Colloquium Mathematicum

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2008

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tom 110

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nr 1

201-204

EN

We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X̃,𝓑_{X̃},m)$. Consider the associated unitary operators on $L²(X̃,𝓑_{X̃},m)$ given by $Ũ_{S}f = √(d(m∘ S)/dm) · (f∘S)$ and $φ·Ũ_{S}$, where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

3

On symmetric logarithm and some old examples in smooth ergodic theory

63%

Frączek K., Lemańczyk M.

Fundamenta Mathematicae

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2003

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tom 180

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nr 3

241-255

EN

We give a positive answer to the problem of existence of smooth weakly mixing but not mixing flows on some surfaces. More precisely, on each compact connected surface whose Euler characteristic is even and negative we construct smooth weakly mixing flows which are disjoint in the sense of Furstenberg from all mixing flows and from all Gaussian flows.

4

Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings

51%

Derriennic Y., Frączek K., Lemańczyk M., Parreau F.

Colloquium Mathematicum

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2008

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tom 110

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nr 1

81-115

EN

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined by stationary ergodic symmetric α-stable processes are shown to belong to the ELF class.

Ograniczanie wyników

2 Colloquium Mathematicum

1 Fundamenta Mathematicae

1 Studia Mathematica

4 Frączek K.

2 Lemańczyk M.

1 Derriennic Y.

1 Parreau F.

1 Wysokińska M.

2 2008

1 2003

1 2001

Some examples of cocycles with simple continuous singular spectrum

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

On symmetric logarithm and some old examples in smooth ergodic theory

Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings