Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Ulcigrai, Corinna

doi:10.1017/s0143385706000836

Cited by 41 publications

(91 citation statements)

References 23 publications

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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: 84%

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: 84%

Multiple mixing for a class of conservative surface flows

Fayad

Kanigowski

2015

Invent. math.

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“…Indeed, if a minimal flow has multi-saddles, corresponding to higher-order zeros of η, then ϕ t is mixing, as proved by Kochergin [Koč75]. On the other hand, flows with saddle loops homologous to zero form an open set in the space of multi-valued Hamiltonians, and if there are such saddle loops, then one can typically produce mixing in each minimal component using the mechanism developed in [SK92] for genus one and in [Ulc07b] for higher genus.…”

Section: Definitions and Main Resultsmentioning

confidence: 93%

“…In [Ulc07a] and [Ulc07b] the author proved that special flows over typical IETs under a roof function f having a single asymmetric logarithmic singularity at the origin (i.e., as in Definition 1.1 with s 1 = s 2 = 1 and C + 0 = C − 0 ) are mixing. The same techniques can be applied to the situation 6 The definition of weak mixing was recalled in footnote 2 (page 1745).…”

Section: Interval Exchange Transformations Letmentioning

confidence: 99%

“…Stretching of Birkhoff sums is the main mechanism which produces mixing in special flows over rotations or over interval exchange transformations when the roof function has logarithmic asymmetric singularities (see e.g. [SK92], [Ulc07b]). Lemma 3.1 shows that stretching of the Birkhoff sums is also a necessary condition to produce mixing, when there is rigidity in the base.…”

Section: Lengths-cocycle Let Us Consider the Zorich Mapmentioning

confidence: 99%

“…A similar principal value phenomenon was used, in the case of rotations, in [SU08]. In presence of an asymmetric singularity, as shown in [Ulc07b], S r k (f 0 ) grows as r k log r k on a set of measure tending to 1 as k tends to infinity. 4.1.…”

Section: Remark That Ifmentioning

confidence: 99%

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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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Spectral Theory of Dynamical Systems

Lemańczyk

2009

Encyclopedia of Complexity and Systems Science

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Mixing of asymmetric logarithmic suspension flows over interval exchange transformations

Cited by 41 publications

References 23 publications

Multiple mixing for a class of conservative surface flows

Multiple mixing for a class of conservative surface flows

Absence of mixing in area-preserving flows on surfaces

Spectral Theory of Dynamical Systems

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