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  • 1. Bjerklöv, Kristian
    et al.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universal asymptotics in hyperbolicity breakdown2008In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 21, no 3, p. 557-586Article in journal (Refereed)
    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. Fayad, B. R.
    et al.
    Saprykina, Maria.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Windsor, A.
    Non-standard smooth realizations of Liouville rotations2007In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 27, p. 1803-1818Article in journal (Refereed)
    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.

  • 3. 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.

  • 4. Fayad, B.
    et al.
    Saprykina, Maria.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary2005In: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, E-ISSN 1873-2151, Vol. 38, no 3, p. 339-364Article in journal (Refereed)
    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.

  • 5. Kaloshin, V.
    et al.
    Saprykina, Maria.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits2006In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 15, no 2, p. 611-640Article in journal (Refereed)
    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].

  • 6. 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.

  • 7. 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.

  • 8.
    Saprykina, Maria
    KTH, Superseded Departments, Mathematics.
    Analytic nonlinearizable uniquely ergodic diffeomorphisms on T-22003In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 23, p. 935-955Article in journal (Refereed)
    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.

  • 9.
    Saprykina, Maria
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Domain of analyticity of normalizing transformations2006In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 19, no 7, p. 1581-1599Article in journal (Refereed)
    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).

  • 10.
    Saprykina, Maria
    KTH, Superseded Departments, Mathematics.
    Non-linearizability, unique ergodicity and weak mixing in dynamics2003Doctoral thesis, comprehensive summary (Other scientific)
  • 11.
    Saprykina, Maria
    et al.
    KTH, Superseded Departments, Mathematics.
    Fayad, B
    Weakly mixing diffeomorphisms on the torus, annulus and discArticle in journal (Other academic)
1 - 11 of 11
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