Non-degenerate fixed points and mixing in flows on a 2-torus

Кочергин, Андрей Васильевич

doi:10.1070/sm2003v194n08abeh000762

Cited by 14 publications

(17 citation statements)

References 6 publications

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“…There exists a sequence α r such that α r → 0 as r → ∞ and for all x distinct from singularities of S r (f ), [16]. The same proof applies also for uniquely ergodic IETs.…”

Section: Mixing Of Asymmetric Logarithmic Suspension Flows Over Iets mentioning

confidence: 72%

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Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Ulcigrai¹

2007

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385706000836How to cite this article: CORINNA ULCIGRAI (2007). Mixing of asymmetric logarithmic suspension ows over interval exchange transformations.Abstract. We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

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“…There exists a sequence α r such that α r → 0 as r → ∞ and for all x distinct from singularities of S r (f ), [16]. The same proof applies also for uniquely ergodic IETs.…”

Section: Mixing Of Asymmetric Logarithmic Suspension Flows Over Iets mentioning

confidence: 72%

“…The following lemma is proved by Kochergin in [16] as Lemma 5.1. The following lemma is proved by Kochergin in [16] as Lemma 5.1.…”

Section: Mixing Of Asymmetric Logarithmic Suspension Flows Over Iets mentioning

confidence: 97%

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Ulcigrai¹

2007

Ergod. Th. Dynam. Sys.

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“…The latter is controlled by the Birkhoff sums of the derivative of the ceiling function that were the subject of investigation of several other works (see e.g. [8,17,19,20,32,35]). …”

Section: Outline Of the Proofmentioning

confidence: 99%

“…The study of the mixing properties of surface flows has known a revival of interest since the beginning of the 2000s, with results such as the computation of the speed of mixing [6] or extensions of the Khanin-Sinai mixing result to include all irrational translation vectors [19,20] (see also [21]), or advances in the study of multi-valued Hamiltonian flows on surfaces in the general case where the Poincaré section return map is an IET and not just a circular rotation [3,[35][36][37].…”

Section: Introductionmentioning

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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“…The diophantine condition of [SK92] was weakened by Kochergin in a series of works ([Koč03], [Koč04a], [Koč04b], [Koč04c]).…”

Section: Interval Exchange Transformations Letmentioning

confidence: 99%

Absence of mixing in area-preserving flows on surfaces

Ulcigrai

2011

Ann. Math.

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We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus g ≥ 2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. Definitions and main results1.1. Flows given by multi-valued Hamiltonians. Let us consider the following natural construction of area-preserving flows on surfaces. On a closed connected orientable surface S of genus g ≥ 1 with a fixed smooth area form ω, consider a smooth closed real-valued differential 1-form η. Let X be the vector field determined by η = i X ω = ω(η, ·) and consider the flow {ϕ t } t∈R on S associated to X. Since η is closed, the transformations ϕ t , t ∈ R, are areapreserving. The flow {ϕ t } t∈R is known as the multi-valued Hamiltonian flow associated to η. Indeed, the flow {ϕ t } t∈R is locally Hamiltonian; i.e., locally one can find coordinates (x, y) on S in which it is given by the solution to the equationsẋ = ∂H/∂y,ẏ = −∂H/∂x for some smooth real-valued Hamiltonian function H. A global Hamiltonian H cannot be in general be defined (see [NZ99, §1.3.4]), but one can think of {ϕ t } t∈R as globally given by a multi-valued Hamiltonian function.The study of flows given by multi-valued Hamiltonians was initiated by S. P. Novikov [Nov82] in connection with problems arising in solid-state physics i.e., the motion of an electron in a metal under the action of a magnetic field. The orbits of such flows arise also in pseudo-periodic topology, as hyperplane sections of periodic surfaces in T n (see e.g. Zorich [Zor99]).From the point of view of topological dynamics, a decomposition into minimal components (i.e., subsurfaces on which the flow is minimal) and periodic components on which all orbits are periodic (elliptic islands around a center and cylinders filled by periodic orbits) was proved independently by Maier

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Non-degenerate fixed points and mixing in flows on a 2-torus

Cited by 14 publications

References 6 publications

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Multiple mixing for a class of conservative surface flows

Absence of mixing in area-preserving flows on surfaces

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