Analytic mixing reparametrizations of irrational flows

Fayad, Bassam

doi:10.1017/s0143385702000214

Cited by 54 publications

(93 citation statements)

References 7 publications

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“…Precisely, we want to see a macroscopic yet controlled drift between the Birkhoff sums of two points 1/(q s ln q s ) d(x, y) 1/(q s+1 ln q s+1 ) for every s sufficiently large. We explain now the proof showing also how the argument simplifies if K α = {n ∈ N : q s+1 < q s (ln q s ) 7 8 } contains all the integers after some n 0 , in particular if α is of constant type.…”

Section: Outline Of the Proofmentioning

confidence: 99%

Multiple mixing for a class of conservative surface flows

Fayad

Kanigowski

2015

Invent. math.

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Arnold and Kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established, nor disproved. Under suitable arithmetic conditions on their unique rotation vector, of full Lebesgue measure in the first case and of full Hausdorff dimension in the second, we show that these flows are mixing of any order. For this, we show that they display a generalization of the so called Ratner property on slow divergence of nearby orbits, that implies strong restrictions on their joinings, which in turn yields higher order mixing. This is the first case in which the Ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications.

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Section: Outline Of the Proofmentioning

confidence: 99%

Multiple mixing for a class of conservative surface flows

Fayad

Kanigowski

2015

Invent. math.

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“…For n ≥ 3, (3.3) does not hold. In relation to this remark, we mention the following important result from [24]: There exist incommensurable numbers α 1 , . .…”

Section: Diffusion and Limit Mixingmentioning

confidence: 94%

Dynamical systems with multivalued integrals on a torus

Козлов

2007

Proc. Steklov Inst. Math.

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Abstract-Properties of the solutions to differential equations on the torus with a complete set of multivalued first integrals are considered, including the existence of an invariant measure, the averaging principle, and the infiniteness of the number of zeros for integrals of zero-mean functions along trajectories. The behavior of systems with closed trajectories of large period is studied. It is shown that a generic system acquires a limit mixing property as the periods tend to infinity.

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“…Constructions of this kind appear most naturally when the resulting diffeomorphism is constructed to be measure-theoretically conjugate to a map of a particular kind, but they also appear when one constructs transformations with more than one ergodic component [157]. This category also includes mixing constructions which were first introduced for time changes for flows on higher-dimensional tori [43,44] and were developed in [47,Section 6] in the context of the approximation by conjugation method. In the latter case one needs to start from a smooth action of a torus rather than of a circle.…”

Section: Non-generic Constructionsmentioning

confidence: 99%