On the cohomological equation for nilflows

Flaminio, Livio; Forni, Giovanni

doi:10.3934/jmd.2007.1.37

Cited by 19 publications

(12 citation statements)

References 14 publications

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“…As in the case of pseudo-Anosov diffeomorphisms, the key step is to characterize iterated coboundaries. Coboundaries were characterized in [FF06] and [FF07] (see also [F14]).…”

Section: Ruelle Resonances For (Partially Hyperbolic) Heisenberg Auto...mentioning

confidence: 99%

Ruelle resonances from cohomological equations

Forni

2020

Preprint

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These notes are based on lectures given by the author at the Summer School on Teichmüller dynamics, mapping class groups and applications in Grenoble, France, in June 2018 and at the Oberwolfach Seminar on Anisotropic Spaces and their Applications to Hyperbolic and Parabolic Systems in June 2019. We derive results about the so-called Ruelle resonances and the asymptotics of correlations for several classes of systems from known results on cohomological equations and invariant distributions for the respective unstable vector fields. In particular, we consider pseudo-Anosov diffeomorphisms on surfaces of higher genus, for horocycle flows on surfaces of constant negative curvature and for partially hyperbolic automorphisms of Heisenberg 3dimensional nilmanfolds. Ruelles resonances for pseudo-Anosov maps with applications to the cohomological equation for their unstable translation flows was recently studied in depth by F. Faure, S. Gouëzel and E. Lanneau [FGL] by methods based on the analysis of the transfer operator of the pseudo-Anosov map. Ruelle resonances for geodesic flows on hyperbolic compact manifolds of any dimension and of partially hyperbolic automorphisms of Heisenberg 3-dimensional nilmanfolds are studied by general results of Dyatlov, Faure and Guillarmou [DFG] and Faure and Tsujii [FT15] based on methods of semi-classical analysis. These works do not derive results on cohomological equations for unstable flows or horospherical foliations of these systems.

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“…As in the case of pseudo-Anosov diffeomorphisms, the key step is to characterize iterated coboundaries. Coboundaries were characterized in [FF06] and [FF07] (see also [F14]).…”

Section: Ruelle Resonances For (Partially Hyperbolic) Heisenberg Auto...mentioning

confidence: 99%

Ruelle resonances from cohomological equations

Forni

2020

Preprint

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“…3.10) exploit that small segments of geodesics curves, pushed by the horocycle flow, are sheared in the horocycle direction. 24 Furthermore, since this is essentially a geometric mechanism for explaining mixing, this phenomenon persists under perturbation and hence can be used also for time-changes (see [29,61], where we prove quantitative mixing results and show polynomial estimates on the decay of correlations for smooth time-changes of the horocycle flow). A similar mechanism, namely shearing of segments of a suitable foliation (but with the difference that the direction of shearing is not global but depends on the segment considered) was also exploited in [20] to prove mixing in some (exceptional) elliptic flows 25 and in the context of nilflows: while nilflows are never mixing (see footnote 14), in suitable classes of smooth time-changes one can implement this mechanism to prove mixing using shearing, see [4,5,74].…”

Section: The Role Of Shearing In Slow Mixingmentioning

confidence: 73%

“…Let us also remark that finitely many invariant distributions for horocycle flows in the finite area, non-compact case were first constructed by Sarnak [76] by methods based on Eisenstein series. The structure of the space of obstructions was described in the case of translation flows (and locally Hamiltonian flows on surfaces) in [27], for nilflows in [24] and for horocycle flows in [22].…”

Section: Chaotic Propertiesmentioning

confidence: 99%

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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“…A result on the existence of solutions of the cohomological equation for twisted horocycle flows was recently proved by Flaminio,Forni and Tanis [FFT16], who were motivated by applications to the cohomological equation for horocycle time-τ maps (see also [Ta12]) and to deviation of ergodic averages for twisted horocycle integrals and horocycle time-τ maps. Twisted nilflows are still nilflows, hence the theory of twisted cohomological equations in the nilpotent case is covered by the general results of Flaminio and Forni [FlaFo07]. As for results on deviation of ergodic averages for nilflows, they are related to bounds on Weyl sums for polynomials.…”

Section: Introductionmentioning

confidence: 99%

Twisted cohomological equations for translation flows

Forni

2021

Ergod. Th. Dynam. Sys.

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We prove by methods of harmonic analysis a result on the existence of solutions for twisted cohomological equations on translation surfaces with loss of derivatives at most $3+$ in Sobolev spaces. As a consequence we prove that product translation flows on (three-dimensional) translation manifolds which are products of a (higher-genus) translation surface with a (flat) circle are stable in the sense of A. Katok. In turn, our result on product flows implies a stability result of time- $\tau $ maps of translation flows on translation surfaces.

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On the cohomological equation for nilflows

Cited by 19 publications

References 14 publications

Ruelle resonances from cohomological equations

Ruelle resonances from cohomological equations

Slow chaos in surface flows

Twisted cohomological equations for translation flows

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