Nondegenerate saddle points and the absence of mixing in flows on surfaces

Кочергин, Андрей Васильевич

doi:10.1134/s0081543807010130

Cited by 13 publications

(10 citation statements)

References 5 publications

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“…3.2) is devised to deal with flows which display absence of mixing. An important early criterion for absence of mixing appears in Katok's work [Ka80], which shows that special flows over IETs under roof functions of bounded variation are never mixing, and by Kochergin's, which shows the absence of mixing for special flows over rotations under a roof with a symmetric logarithmic singularity (see [Ko72,Ko07]). Both criteria require as input tightness of Birkhoff sums along some subsequences of rigidity (or partial rigidity) times, i.e.…”

Section: Strategy Of the Proof Of The Main Resultmentioning

confidence: 99%

“…In this case, mixing depends on the (a)symmetry of the singularities. The first result on absence of mixing for special flows with symmetric logarithmic singularities over rotations is due to Kochergin [Ko72] (see also [Ko07] where the result was proved for all irrational frequencies). If the roof has an asymmetric logarithmic singularities, instead, mixing is typical, as it was proved by Sinai-Khanin for a full measure set of rotations numbers (see also further works by Kochergin [Ko75,Ko03,Ko04,Ko04']).…”

Section: Previous Results On Ergodic and Spectral Propertiesmentioning

confidence: 99%

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Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Chaika

Frączek

Kanigowski

et al. 2020

Commun. Math. Phys.

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We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non-degenerate isomorphic saddles has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $$\mathcal {H}(1,1)$$ H ( 1 , 1 ) , yields rigidity sets needed for the singularity result.

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Section: Strategy Of the Proof Of The Main Resultmentioning

confidence: 99%

Section: Previous Results On Ergodic and Spectral Propertiesmentioning

confidence: 99%

Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Chaika

Frączek

Kanigowski

et al. 2020

Commun. Math. Phys.

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“…22 Absence of mixing for typical flows for any g ≥ 2 was proved in [86], while weak mixing is proved also for minimal components of locally Hamiltonian flow with simple saddles in [85]. Let us remark that a result in this direction for g = 1 was already proved by Kocergin in [52] (see also [54]) (in the language of special flows over rotations, which does not have a direct implication for locally Hamiltonian flows but suggested that the absence of mixing could hold also when rotations are replaced by IETs and hence in higher genus. Absence of mixing in the special case of S with g = 2 and locally Hamiltonian flows with two isomorphic simple saddles was shown by Scheglov [78].…”

Section: The Role Of Shearing In Slow Mixingmentioning

confidence: 87%

Slow chaos in surface flows

Ulcigrai¹

2020

Boll Unione Mat Ital

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Flows on surfaces describe many systems of physical origin and are one of the most fundamental examples of dynamical systems, studied since Poincará. In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth area-preserving flows (a class which include locally Hamiltonian flows), thanks to the connection to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics. We motivate and survey some of the recent breakthroughs on their mixing and spectral properties and the mechanisms, such as shearing, on which they are based, which exploit analytic, arithmetic and geometric techniques.

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“…Flows on 2-tori with symmetric logarithmic singularities of return time (e.g., Eq. 6 at f > f crit ) do not mix as well (23). Flows on 2-tori with asymmetric logarithmic singularities of return time (24) and the flows on 2-tori with power-law singularities of return time (25) mix.…”

Section: Singularities Of Passage Timementioning

confidence: 99%

Subdiffusive and superdiffusive transport in plane steady viscous flows

Zaks

Nepomnyashchy

2018

Proc. Natl. Acad. Sci. U.S.A.

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Deterministic transport of passive tracers in steady laminar plane flows of incompressible viscous fluids through lattices of solid bodies or arrays of steady vortices can be anomalous. Motion along regular patterns of streamlines is often aperiodic: Repeated slow passages near stagnation points and/or solid surfaces serve for eventual decorrelation. Singularities of passage times near the obstacles, dictated by the boundary conditions, affect the character of transport anomalies: Flows past arrays of vortices are subdiffusive whereas tracers advected through lattices of solid obstacles can feature superdiffusion. We calculate the transport characteristics with the help of the simple and computationally efficient model: the special flow.

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Nondegenerate saddle points and the absence of mixing in flows on surfaces

Cited by 13 publications

References 5 publications

Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Slow chaos in surface flows

Subdiffusive and superdiffusive transport in plane steady viscous flows

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