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  • 1.
    De la Llave, Rafael
    et al.
    Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA..
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation2022In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, no 3, p. 1166-1187Article in journal (Refereed)
    Abstract [en]

    Consider an analytic Hamiltonian system near its analytic invariant torus T-0 carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at T-0 is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation-not just a formal power series-bringing the Hamiltonian into its Birkhoff normal form.

  • 2.
    de la Llave, Rafael
    et al.
    School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332-1160, USA, 686 Cherry St.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Noncommutative coboundary equations over integrable systems2023In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 19, p. 773-794Article in journal (Refereed)
    Abstract [en]

    We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra (Formula Presented) or a Lie group. Namely, we consider an integrable dynamical system (Formula Presented), and a real-analytic family of cocycles (Formula Presented) indexed by a complex parameter ε in an open ball (Formula Presented). We show that if ηε is close to identity and has trivial periodic data, i.e., (Formula Presented) for each periodic point p = fn p and each (Formula Presented), then there exists a real-analytic family of maps (Formula Presented) satisfying the coboundary equation (Formula Presented) for all (Formula Presented) and (Formula Presented). We also show that if the coboundary equation above with an analytic left-hand side ηε has a solution in the sense of formal power series in ε, then it has an analytic solution.

  • 3.
    Dolgopyat, Dmitry
    et al.
    Univ Maryland, College Pk, MD 20742 USA..
    Fayad, Bassam
    CNRS UMR7586 IMJ PRG Paris, Paris, France..
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment2021In: Electronic Journal of Probability, E-ISSN 1083-6489, Vol. 26, article id 66Article in journal (Refereed)
    Abstract [en]

    We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.

  • 4. Fayad, B.
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Isolated elliptic fixed points for smooth Hamiltonians2017In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 692, p. 67-82Article in journal (Refereed)
    Abstract [en]

    We construct on ℝ2d, for any d ≥ 3, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For d ≥ 4, the Hamiltonians we construct have not any invariant torus of dimension d. Our examples are obtained by a version of the successive conjugation scheme à la Anosov-Katok.

  • 5. Fayad, B.
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Realizing Arbitrary D-Dimensional Dynamics By Renormalization Of Cd-Perturbations Of Identity2022In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 42, no 2, p. 597-604Article in journal (Refereed)
    Abstract [en]

    Any Cd conservative map f of the d-dimensional unit ball Bd, d ≥ 2, can be realized by renormalized iteration of a Cd perturbation of identity: there exists a conservative diffeomorphism of Bd, arbitrarily close to identity in the Cd topology, that has a periodic disc on which the return dynamics after a Cd change of coordinates is exactly f. 

  • 6.
    Fayad, Bassam
    et al.
    IMJ PRG CNRS, UP7D, 58-56 Ave France,Boite Courrier 7012, F-75205 Paris 13, France..
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Topological weak mixing and diffusion at all times for a class of Hamiltonian systems2022In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, no 2, p. 777-791, article id PII S0143385721000122Article in journal (Refereed)
    Abstract [en]

    We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.

  • 7. Kaloshin, Vadim
    et al.
    Levi, Mark
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Arnol ' d Diffusion in a Pendulum Lattice2014In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 67, no 5, p. 748-775Article in journal (Refereed)
    Abstract [en]

    The main model studied in this paper is a lattice of pendula with a nearest-neighbor coupling. If the coupling is weak, then the system is near-integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way.

  • 8. Kaloshin, Vadim
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension2012In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 315, no 3, p. 643-697Article in journal (Refereed)
    Abstract [en]

    The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462-465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797-808, 1998) proposed to look for an example of a Hamiltonian near with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.

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