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On ergodicity of some cylinder flows

Krzysztof Frączek (2000)

Fundamenta Mathematicae

We study ergodicity of cylinder flows of the form   , , where is a measurable cocycle with zero integral. We show a new class of smooth ergodic cocycles. Let k be a natural number and let f be a function such that is piecewise absolutely continuous (but not continuous) with zero sum of jumps. We show that if the points of discontinuity of have some good properties, then is ergodic. Moreover, there exists such that if is a function with zero integral such that is of bounded variation...

On measure theoretical analogues of the Takesaki structure theorem for type III factors

Alexandre Danilenko, Toshihiro Hamachi (2000)

Colloquium Mathematicae

The orbit equivalence of type ergodic equivalence relations is considered. We show that it is equivalent to the outer conjugacy problem for the natural trace-scaling action of a countable dense ℝ-subgroup by automorphisms of the Radon-Nikodym skew product extensions of these relations. A similar result holds for the weak equivalence of arbitrary type cocycles with values in Abelian groups.

On solvability of the cohomology equation in function spaces

Ryotaro Sato (2003)

Studia Mathematica

Let T be an endomorphism of a probability measure space (Ω,𝓐,μ), and f be a real-valued measurable function on Ω. We consider the cohomology equation f = h ∘ T - h. Conditions for the existence of real-valued measurable solutions h in some function spaces are deduced. The results obtained generalize and improve a recent result of Alonso, Hong and Obaya.

On subrelations of ergodic measured type III equivalence relations

Alexandre Danilenko (2000)

Colloquium Mathematicae

We discuss the classification up to orbit equivalence of inclusions 𝑆 ⊂ ℛ of measured ergodic discrete hyperfinite equivalence relations. In the case of type III relations, the orbit equivalence classes of such inclusions of finite index are completely classified in terms of triplets consisting of a transitive permutation group G on a finite set (whose cardinality is the index of 𝑆 ⊂ ℛ), an ergodic nonsingular ℝ-flow V and a homomorphism of G to the centralizer of V.

On the ergodic decomposition for a cocycle

Jean-Pierre Conze, Albert Raugi (2009)

Colloquium Mathematicae

Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure . We consider the map defined on X × G by and the cocycle generated by φ. Using a characterization of the ergodic invariant measures for , we give the form of the ergodic decomposition of or more generally of the -invariant measures , where is χ∘φ-conformal for an exponential χ on G.

Opening gaps in the spectrum of strictly ergodic Schrödinger operators

Artur Avila, Jairo Bochi, David Damanik (2012)

Journal of the European Mathematical Society

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...

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