Nonsingular saddle points and the absence of mixing

Кочергин, Андрей Васильевич

doi:10.1007/bf01437864

Cited by 30 publications

(24 citation statements)

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“…That mixing was indeed possible in the latter context was proved in a particular case by Khanin and Sinai [32]. Absence of mixing in the case of Morse forms was obtained by Kochergin in some particular cases [18] (see also [22]), and later generalized to a typical one form by Ulcigrai [35].…”

Section: Introductionmentioning

confidence: 82%

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: Introductionmentioning

confidence: 82%

Multiple mixing for a class of conservative surface flows

Fayad

Kanigowski

2015

Invent. math.

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“…Note that, as follows from [3], if the invariant measure on T 2 has a smooth positive density, then the singularities of a "roof" function of the special flow are strictly logarithmic.…”

Section: To the Anniversary Of Anatole Katok My Friend And Teachermentioning

confidence: 99%

“…V. I. Arnold has shown that there exists a smooth closed curve transversal to the orbits of the ergodic component. The invariant measure and the flow naturally induce a smooth parameterization on the curve (this procedure was described in detail, e.g., in [3]). The first-return map is determined everywhere on the curve, except for a finite number of points that are the points of the last intersection of the stable separatrices with the curve.…”

Section: To the Anniversary Of Anatole Katok My Friend And Teachermentioning

confidence: 99%

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Well-approximable angles and mixing for flows on 𝕋² with nonsingular fixed points

Кочергин¹

2004

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We consider special flows over circle rotations with an asymmetric function having logarithmic singularities. If some expressions containing singularity coefficients are different from any negative integer, then there exists a class of well-approximable angles of rotation such that the special flow over the rotation of this class is mixing.Examples of smooth flows on a two-dimensional torus with a smooth invariant measure and nonsingular hyperbolic fixed points appear naturally in Arnold's paper [1]. The phase space of such a flow decomposes into cells bounded by closed separatrices of regular fixed points and filled with periodic orbits, and an ergodic component in which orbits on one side of a fixed point visit its neighborhood more frequently than on the other. (See the figure, for example.) V. I. Arnold has shown that there exists a smooth closed curve transversal to the orbits of the ergodic component. The invariant measure and the flow naturally induce a smooth parameterization on the curve (this procedure was described in detail, e.g., in [3]). The first-return map is determined everywhere on the curve, except for a finite number of points that are the points of the last intersection of the stable separatrices with the curve. In the induced parameterization, this map is a circle rotation. The return time is a smooth function of the parameter everywhere

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“…On the other hand, if the section map happens to be a rotation then any flow with only non-degenerate saddles is not mixing [92].…”

Section: Flows On Surfaces Of Higher Genusmentioning

confidence: 99%