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  • 1. Bjerklöv, Kristian
    et al.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Universal asymptotics in hyperbolicity breakdown2008Inngår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 21, nr 3, s. 557-586Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study a scenario for the disappearance of hyperbolicity of invariant tori in a class of quasi-periodic systems. In this scenario, the system loses hyperbolicity because two invariant directions come close to each other, losing their regularity. In a recent paper, based on numerical results, Haro and de la Llave (2006 Chaos 16 013120) discovered a quantitative universality in this scenario, namely, that the minimal angle between the two invariant directions has a power law dependence on the parameters and the exponents of the power law are universal. We present an analytic proof of this result.

  • 2.
    De la Llave, Rafael
    et al.
    Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA..
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation2022Inngår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, nr 3, s. 1166-1187Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 3.
    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, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Noncommutative coboundary equations over integrable systems2023Inngår i: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 19, s. 773-794Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 4.
    Dolgopyat, Dmitry
    et al.
    Univ Maryland, College Pk, MD 20742 USA..
    Fayad, Bassam
    CNRS UMR7586 IMJ PRG Paris, Paris, France..
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment2021Inngår i: Electronic Journal of Probability, E-ISSN 1083-6489, Vol. 26, artikkel-id 66Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 5. Fayad, B. R.
    et al.
    Saprykina, Maria.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Windsor, A.
    Non-standard smooth realizations of Liouville rotations2007Inngår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 27, s. 1803-1818Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We augment the C-infinity conjugation approximation method with explicit estimates on the conjugacy map. This allows us to construct ergodic volume-preserving diffeomorphisms measure-theoretically isomorphic to any a priori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic.

  • 6. Fayad, B.
    et al.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Isolated elliptic fixed points for smooth Hamiltonians2017Inngår i: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 692, s. 67-82Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 7. Fayad, B.
    et al.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Realizing Arbitrary D-Dimensional Dynamics By Renormalization Of Cd-Perturbations Of Identity2022Inngår i: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 42, nr 2, s. 597-604Artikkel i tidsskrift (Fagfellevurdert)
    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. 

  • 8. Fayad, B.
    et al.
    Saprykina, Maria.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary2005Inngår i: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, E-ISSN 1873-2151, Vol. 38, nr 3, s. 339-364Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let M be an m-dimensional differentiable manifold with a nontrivial circle action S = {S-t}(t is an element of R), St+1 = S-t, preserving a smooth volume mu. For any Liouville number alpha we construct a sequence of area-preserving diffeomorphisms H-n such that the sequence H-n circle S-alpha circle H-n(-1) converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1]. For m = 2 and M equal to the unit disc D-2 = {x(2) + y(2) <= 1} or the closed annulus A = T x [0, 1] this result proves the following dichotomy: alpha is an element of R \ Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals alpha (on at least one of the boundaries in the case of A). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if alpha is Diophantine, then any area preserving diffeomorphism with rotation number alpha on the boundary (on at least one of the boundaries in the case of A) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.

  • 9.
    Fayad, Bassam
    et al.
    IMJ PRG CNRS, UP7D, 58-56 Ave France,Boite Courrier 7012, F-75205 Paris 13, France..
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Topological weak mixing and diffusion at all times for a class of Hamiltonian systems2022Inngår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, nr 2, s. 777-791, artikkel-id PII S0143385721000122Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 10. Kaloshin, V.
    et al.
    Saprykina, Maria.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits2006Inngår i: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 15, nr 2, s. 611-640Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let M be a compact manifold of dimension three with a nondegenerate volume form Omega and Diff(Omega)(r) (M) be the space of C-r-smooth (Omega-) volume-preserving difffeomorphisms of M with 2 <= r <= infinity. In this paper we prove two results. One of them provides the existence of a Newhouse domain N in Diff(Omega)(r)(M). The proof is based on the theory of normal forms [13], construction of certain renormalization limits, and results from [23, 26, 28, 32]. To formulate the second one, associate to each diffeomorphism a sequence P-n(f) which gives for each n the number of isolated periodic points of f of period n. The main result of this paper states that for a Baire generic diffeomorphism f in N, the number of periodic points P-n(f) grows with n faster than any prescribed sequence of numbers {a(n)} (n is an element of Z+) along a subsequence, i.e., P-ni (f) > ani for some n(i) -> with infinity i -> infinity. The strategy of the proof is similar to the one of the corresponding 2-dimensional non volume-preserving result [16]. The latter one is, in its turn, based on the Gonchenko-Shilnikov-Turaev Theorem [8, 9].

  • 11. Kaloshin, Vadim
    et al.
    Levi, Mark
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Arnol ' d Diffusion in a Pendulum Lattice2014Inngår i: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 67, nr 5, s. 748-775Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 12. Kaloshin, Vadim
    et al.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension2012Inngår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 315, nr 3, s. 643-697Artikkel i tidsskrift (Fagfellevurdert)
    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.

  • 13.
    Saprykina, Maria
    KTH, Tidigare Institutioner (före 2005), Matematik.
    Analytic nonlinearizable uniquely ergodic diffeomorphisms on T-22003Inngår i: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 23, s. 935-955Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper we study the behavior of diffeomorphisms, contained in the closure (A) over bar (alpha) (in the inductive limit topology) of the set A(alpha) of real-analytic diffeomorphisms of the torus T-2, which are conjugated to the rotation R-alpha : (x, y) hooked right arrow (x+alpha, y) by an analytic measure-preserving transformation. We show that for a generic alpha is an element of [0, 1], (A) over bar (alpha) contains a dense set of uniquely ergodic diffeomorphisms. We also prove that (A) over bar (alpha) contains a dense set of diffeomorphisms that are minimal and non-ergodic.

  • 14.
    Saprykina, Maria
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Domain of analyticity of normalizing transformations2006Inngår i: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 19, nr 7, s. 1581-1599Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We investigate questions of divergence or local convergence of (formal) normalizing transformations associated with the Birkhoff normal form (BNF) at the origin of a holomorphic Hamiltonian system. These questions are addressed for systems for which the BNF is a quadratic function H-Lambda = Sigma(d)(j=1) lambda(j) x(j) y(j), Lambda := (lambda(1),..., lambda(d)) being a non-resonant, either real or purely imaginary, vector. We prove that for a generic Lambda is an element of R-d or i Lambda is an element of R-d one can define Hamiltonians H = H-Lambda + (H) over cap satisfying the following properties: (i) H is real-analytic, holomorphic in the unit polydisc D(1), and H is defined arbitrarily close to H-Lambda, (ii) the BNF of H equals H-Lambda and (iii) any symplectic normalizing transformation diverges, or given any 0 < rho < 1 any normalizing transformation diverges outside the polydisc of radius rho, and there is a real-analytic normalizing transformation (converging in a smaller domain).

  • 15.
    Saprykina, Maria
    KTH, Tidigare Institutioner (före 2005), Matematik.
    Non-linearizability, unique ergodicity and weak mixing in dynamics2003Doktoravhandling, med artikler (Annet vitenskapelig)
  • 16.
    Saprykina, Maria
    et al.
    KTH, Tidigare Institutioner (före 2005), Matematik.
    Fayad, B
    Weakly mixing diffeomorphisms on the torus, annulus and discArtikkel i tidsskrift (Annet vitenskapelig)
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