Multiple mixing for a class of conservative surface flows

Fayad, Bassam; Kanigowski, Adam

doi:10.1007/s00222-015-0596-6

Cited by 33 publications

(77 citation statements)

References 52 publications

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“…It follows that going either forward or backward we can always recede from the dangerous interval around 0. A complete proof of this statement is given in [5] Lemma 4.6.…”

Section: Pd-property For Arnol'd Flows Proof Of Proposition 53mentioning

confidence: 96%

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Slow entropy for some smooth flows on surfaces

Kanigowski

2018

Isr. J. Math.

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We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C 2 everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol'd flows) we show that in the scale a n (t) = n(log n) t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x −γ , γ ∈ (0, 1)) (Kochergin flows) we show that in the scale a n (t) = n t slow entropy equals 1 + γ for a.e. α. We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n) t scale and is at most 1 for scale n t . As a consequence we get that a.e. Arnol'd and a.e Kochergin flow is never of local rank one.

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“…It follows that going either forward or backward we can always recede from the dangerous interval around 0. A complete proof of this statement is given in [5] Lemma 4.6.…”

Section: Pd-property For Arnol'd Flows Proof Of Proposition 53mentioning

confidence: 96%

“…It is shown in [5] that λ(E) = 1. With the above definitions our main theorems are the following (see Section 2 for the precise definition of h β s ):…”

Section: Introductionmentioning

confidence: 99%

Slow entropy for some smooth flows on surfaces

Kanigowski

2018

Isr. J. Math.

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“…All the variants of the Ratner properties are defined so that, if (T t ) has the SR-property, then it has the finite extension of joinings property (shortened as FEJ property, see [10]), which is a rigidity property that restricts the type of self-joinings that (T t ) can have (see [33,15]).…”

Section: Ratner Propertiesmentioning

confidence: 99%

“…The set Z is such that the points y, y satisfy a Ratner-type form of shearing either going forward or backward in time (see (7.4)). This is essentially the set on which the switchable Ratner property (see Section 2.5) holds for the Arnold flow and the definition is indeed the same as the set Z is in [10] or [21]. If y, y display this good form of Ratner-like shearing going forward, we then show that the Forward assumptions (F 1) − (F 4) in F. of Proposition 5.1 hold, while if the Ratner-like form of shearing happens backward, we show that the Backward assumptions (B1) − (B4) in B. of Proposition 5.1 hold.…”

Section: Disjointness In Arnol'd Flows (Proof Of Theorem 12)mentioning

confidence: 99%

On disjointness properties of some parabolic flows

2020

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The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed forward the disjointness theory of such systems. In this paper, exploiting a recent variation of the Ratner property, we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world. In particular, we establish a general disjointness criterion based on the switchable Ratner property. We then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth Arnol'd flows on T 2 , focusing in particular on disjointness of distinct flow rescalings. As a consequence, we answer a question by Marina Ratner on the Möbius orthogonality of time-changes of horocycle flows. In fact, we prove Möbius orthogonality for all smooth time-changes of horocycle flows and uniquely ergodic realizations of Arnol'd flows considered. Dedicated to the memory of Marina RatnerContents 49 7.3. Slow shearing properties: proof of Lemma 7.2 51 7.4. Splitting of orbits: proof of Lemma 7.3 53 8. Disjointness of time changes of horocycle flows and Arnol'd flows (proof of Theorem 1.3) 60 9. Time changes of horocycle flows and Sarnak's conjecture (answer to M. Ratner's question) 64 Appendix A. Consequences of shearing for time changes of horocycle flows 66 Appendix B. Ergodic averages for horocycle flows 71 Appendix C. Strong MOMO and USIC properties 73 Acknowledegements 75 References 75 8 ADAM KANIGOWSKI, MARIUSZ LEMAŃCZYK, AND CORINNA ULCIGRAI flows (Theorem 1.3).Finally, in Section 9 we discuss consequences of our main results to the problem of Möbius disjointness. Three appendices follow. In the first two, we give the proofs of two technical results which are used in the proof of Theorem 1.1: in the first one, Appendix A, we prove some quantitative shearing properties for time-changes of horocycle flows; in Appendix B we state a result on ergodic averages of a special family of smooth functions which can be deduced from [6] (we thank G. Forni for the proof). Finally, in Appendix C, we present a missing link in the literature, namely, the equivalence between two different kinds of convergence on short intervals.2. Preliminaries: definitions, notation and some basic facts 2.1. Flows, joinings and disjointness. Assume that (Z, D, κ) is a probability standard Borel space. If Z is additionally a metric space (with a metric d), then we will also write (Z, D, κ, d) to emphasize the role of d. By Aut(Z, D, κ), we denote the set of all automorphisms of (Z, D, κ), i.e. R ∈ Aut(Z, D, κ) if R : Z → Z is a bi-measurable, measure-preserving κ-a.e. bijection. Each R ∈ Aut(Z, D, κ) determines a unitary operator, also denoted by R, on L 2 (Z, D, κ) given by Rf := f • R for f ∈ L 2 (Z, D, κ). Endowed with the weak operator topology of unitary operators, Aut(Z, D, κ) becomes a Polish group. We will be mainly deal with flows, i.e. with measurable, measure-pr...

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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 in full generality. 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, it was shown in [42] that these flows are mixing of any order, [59] for flows on higher genus surfaces).…”

Section: Spectral Type Of Related Systemsmentioning

confidence: 99%

Some Questions Around Quasi-Periodic Dynamics

Fayad¹,

Krikorian²

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems.In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic.The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

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Multiple mixing for a class of conservative surface flows

Cited by 33 publications

References 52 publications

Slow entropy for some smooth flows on surfaces

Slow entropy for some smooth flows on surfaces

On disjointness properties of some parabolic flows

Some Questions Around Quasi-Periodic Dynamics

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