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  • 1.
    Benedicks, Michael
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Misiurewicz, Michal
    Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202, 402 N. Blackford Street, USA .
    Rodrigues, Ana
    Department of Mathematics, Exeter University, Exeter EX4 4QF, UK.
    Expansion properties of double standard maps2023In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 43, no 8, p. 2549-2588Article in journal (Refereed)
    Abstract [en]

    For the family of double standard maps we investigate the structure of the space of parameters a when and when. In the first case the maps have a critical point, but for a set of parameters of positive Lebesgue measure there is an invariant absolutely continuous measure for. In the second case there is an open non-empty set of parameters for which the map is expanding. We show that as, the set accumulates on many points of in a regular way from the measure point of view.

  • 2.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On some generalizations of skew-shifts on T-22019In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 39, p. 19-61Article in journal (Refereed)
    Abstract [en]

    In this paper we investigate maps of the two-torus T-2 of the form T (x, y) = (x + omega, g(x) + f (y)) for Diophantine omega is an element of T and for a class of maps f, g : T -> T, where each g is strictly monotone and of degree 2 and each f is an orientation-preserving circle homeomorphism. For our class of f and g, we show that T is minimal and has exactly two invariant and ergodic Borel probability measures. Moreover, these measures are supported on two T-invariant graphs. One of the graphs is a strange non-chaotic attractor whose basin of attraction consists of (Lebesgue) almost all points in T-2. Only a low-regularity assumption (Lipschitz) is needed on the maps f and g, and the results are robust with respect to Lipschitz-small perturbations of f and g.

  • 3.
    Bjerklöv, Kristian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrodinger equations2005In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 25, p. 1015-1045Article in journal (Refereed)
    Abstract [en]

    We study the discrete quasi-periodic Schrodinger equation -(u(n+1) + u(n-1)) + lambda V(theta + n omega)u(n) = Eu-n with a non-constant C-1 potential function V : T -> R. We prove that for sufficiently large k there is a set Omega subset of T of frequencies omega, whose measure tends to 1 as lambda -> infinity, with the following property. For each w e Q there is a 'large' (in measure) set of energies E, all lying in the spectrum of the associated Schrodinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrodinger cocycle is minimal but not ergodic.

  • 4.
    Damjanovic, Danijela
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Xu, Disheng
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions2020In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 40, no 1, p. 117-141Article in journal (Refereed)
    Abstract [en]

    We prove that every smooth diffeomorphism group valued cocycle over certain Z(k) Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan Z(k) (k >= 3) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over Z(k) actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

  • 5.
    Damjanović, Danijela
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Tanis, J. A. M. E. S.
    Transversal local rigidity of discrete abelian actions on Heisenberg nilmanifolds2021In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, p. 1-41Article in journal (Refereed)
    Abstract [en]

    In this paper we prove a perturbative result for a class of actions on Heisenberg nilmanifolds that have Diophantine properties. Along the way we prove cohomological rigidity and obtain a tame splitting for the cohomology with coefficients in smooth vector fields for such actions.

  • 6.
    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.

  • 7. 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.

  • 8.
    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.

  • 9.
    Karlsson, Anders
    KTH, Superseded Departments (pre-2005), Mathematics.
    Non-expanding maps and Busemann functions2001In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 21, p. 1447-1457Article in journal (Refereed)
    Abstract [en]

    We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally compact metric spaces and generalize the theorems of Wolff and Denjoy about the iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements which can sometimes be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. A few further remarks of similar type are made concerning certain random products of sernicontractions and also concerning semicontractions of Gromov hyperbolic spaces.

  • 10. Martens, M.
    et al.
    Winckler, Björn
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Physical measures for infinitely renormalizable Lorenz maps2018In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 38, p. 717-738Article in journal (Refereed)
    Abstract [en]

    A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics: namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article, we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure is the control of the position of these critical points.

  • 11.
    Miles, Richard
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Finitely represented closed-orbit subdynamics for commuting automorphisms2010In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 30, p. 1787-1802Article in journal (Refereed)
    Abstract [en]

    The purpose of this paper is to exhibit highly structured subdynamics for a class of non-expansive algebraic Z(d)-actions based on the closed orbits of elements of an action. This is done using dynamical Dirichlet series to encode orbit counts. It is shown that there is a distinguished group homomorphism from Z(d) onto a finite abelian group that controls the form of the Dirichlet series of elements of an action and that these series have common analytic properties. Corresponding orbit growth asymptotics are subsequently investigated.

  • 12. Przytycki, F
    et al.
    Rivera-Letelier, J
    Smirnov, Stanislav
    KTH, Superseded Departments (pre-2005), Mathematics.
    Equality of pressures for rational functions2004In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 24, p. 891-914Article in journal (Refereed)
    Abstract [en]

    We prove that for all rational functions f on the Riemann sphere and potential -t ln \f'\ t greater than or equal to 0 all the notions of pressure introduced in Przytycki (Proc. Amer Math. Soc. 351(5) (1999), 2081-2099) coincide. In particular, we get a new simple proof of the equality between the hyperbolic Hausdorff dimension and the minimal exponent of conformal measure on a Julia set. We prove that these pressures are equal to the pressure defined with the use of periodic orbits under an assumption that there are not many periodic orbits with Lyapunov exponent close to 1 moving close together, in particular under the Topological Collet-Eckmann condition. In Appendix A, we discuss the case t < 0.

  • 13.
    Saprykina, Maria
    KTH, Superseded Departments (pre-2005), 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.

  • 14.
    Schnellmann, Daniel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Non-continuous weakly expanding skew-products of quadratic maps with two positive Lyapunov exponents2008In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 28, no 1, p. 245-266Article in journal (Refereed)
    Abstract [en]

    We study an extension of the Viana map where the base dynamics isa discontinuous expanding map, and prove the existence of two positive Lyapunovexponents.

  • 15.
    Thunberg, Hans
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Positive exponent in families with flat critical point1999In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 19, no 3, p. 767-807Article in journal (Refereed)
    Abstract [en]

    It is known that in generic, full unimodal families with a critical point of finite order, there exists a set of positive measure in parameter space such that the corresponding maps have chaotic behaviour. In this paper we prove the corresponding statement for certain families of unimodal maps with flat critical point. One of the key-points is a large deviation argument for sums of ‘almost’ independent random variables with only finitely many moments.

1 - 15 of 15
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