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Ergodic Theory and Dynamical Systems 2012-13

The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute
Organizers: Bram Mesland , Richard Sharp

For the schedule of the mini-courses please visit this page

Term 3

  • April 23, 2013
    Gunther Cornelissen (University of Utrecht)
    Title: Dynamical systems and point counting
    Abstract: Counting the points of a curve over finite fields doesn't determine the curve up to isomorphism; this is analogous to the famous fact that you cannot "hear the shape of a drum". I will show how to remedy this by counting points in a weighted way (namely, using abelian L-series). The underlying structure is a dynamical system that models actions from class field theory, and - curiously - arose from a study of a quantum statistical mechanical system.
  • April 30, 2013
    Uwe Grimm (Open University)
    Title: Singular Continuous Diffraction of the Squiral Tiling
    Abstract: The Thue-Morse system is a paradigm of singular continuous diffraction in one dimension. After a short introduction to mathematical diffraction theory, I shall briefly discuss the Thue-Morse system and some one-dimensional generalisations. As my main example, I am then going to consider in detail a planar system constructed by a bijective block substitution rule, which is equivalent to the inflation rule of the squiral tiling. For balanced weights, the corresponding diffraction measure is purely singular continuous, whereas the dynamical spectrum also contains a pure point component. The diffraction measure is a two-dimensional Riesz product that can be calculated explicitly.
  • May 7, 2013
    Jaap Eldering (Imperial College)
    Title: Normally hyperbolic invariant manifolds: noncompactness and geometry
    Abstract: As generalization of hyperbolic fixed points, normally hyperbolic invariant manifolds (NHIMs for short) are an important tool to globally study perturbations of dynamical systems. In this talk, I will first briefly review the theory of NHIMs and their use in applications. Then I will formulate the theorem on persistence of NHIMs and show how my result generalizes the classical compact to a noncompact differential geometric setting. Noncompactness requires us to introduce the concept of Riemannian manifolds of bounded geometry. These can be viewed as the class of uniformly C^k manifolds; I will illustrate this concept with images. Finally, I will describe why bounded geometry is needed and try to say a few words about how it enters the proof of persistence of NHIMs.
  • May 14, 2013
    Markus Fraczek (University of Warwick)
    Title: Spectra of transfer operators and zeta functions
    Abstract: The transfer operator method is one of the very few methods enabling the evaluation of certain zeta functions, such as the Selberg zeta function and the Ruelle zeta function for the Kac-Baker model. To evaluate these zeta functions we have to determine the spectra of the corresponding transfer operators. Since there is no great hope of an analytical approach to the spectrum of the transfer operator, we study its spectrum mainly using numerical methods. We will construct nuclear representations for certain transfer operator and approximate them by finite matrices, which can be used for numerical computations. Numerical results obtained by implementing this method will be also presented.
  • May 21, 2013
    Charles Walkden (University of Manchester)
    Title: Instances of stochastic stability for generalised IFSs
  • May 28, 2013
    Bram Mesland (University of Warwick)
    Title: Limit sets, Patterson-Sullivan measure and noncommutative geometry
    Abstract: The limit set of a Kleinian group is the locus for complicated dynamics of this group. Canonically associated to the action of the group on hyperbolic n-space is its limit set and the Patterson-Sullivan measure. The properties of this measure at infinity reflect the geometry of the action on hyperbolic n-space. To study the ergodic action on the limit set, we replace the (ill-defined) quotient space by a noncommutative C*-algebra, the crossed product. We show how the underlying hyperbolic geometry can be used to define interesting dynamics on this C*-algebra. This will result in a noncommutive metric structure, or spectral triple, for the crossed product algebra. Such spectral triples canonically induce functionals on the K-theory of the C*-algebra, which in this case, by a result of Manin-Marcolli, is related to the homology of the associated hyperbolic quotient manifold. This is a report on work in progress.
  • June 18, 2013
    Peter Hazard (University of Warwick)
    Title: Renormalisation of Henon-like Maps in Dimension Two
    ​Abstract: I will discuss recent developments in the renormalization theory for Henon-like maps in two dimensions by Lyubich, Martens, de Carvalho and myself. Even in the strongly dissipative case there are surprising differences with the unimodal picture. For example, universal phenomena observed in the one-dimensional unimodal setting for infinitely renormalisable maps persists in the Henon-like setting, but the attractors are no longer rigid. I will discuss this and some other results and, time permitting, mention a few conjectures suggested by numerical experiments.

Term 2

  • January 15, 2013
    Mike Todd (University of St Andrews)
    Title: Dynamical systems with holes: escape rates, equilibrium states and dimension
    Abstract: Given a smooth interval map f:I --> I with one or more critical points, we consider
    the dynamical behaviour of the system when a hole is punctured in the system, allowing
    mass to escape. Work by Bruin, Demers and Melbourne studied the escape of Lebesgue
    measure, obtaining a conditionally invariant measure absolutely continuous w.r.t. Lebesgue,
    as well as a related measure on the survivor set (the set of points which never escape).
    This was related to the study of equilibrium states. In this talk, motivated by Bowen's
    formula for the dimension of dynamically defined sets, I'll consider escape w.r.t. other
    natural measures and thus give an expression for the Hausdorff dimension of the survivor set.
    This is joint work with M. Demers.
  • January 22, 2013
    Sebastian van Strien (Imperial College London)
    Title: Stochastic stability
  • January 29, 2013
    Oleg Kozlovski (University of Warwick)
    Title: Hilbert-Arnold problem for 1D maps
    Abstract: In this talk we will discuss if maps in a "typical" family of maps of an interval or
    a circle have bounded number of attracting trajectories. The answer depends on what
    "typical" means, and on the smoothness of the maps.
  • February 5, 2013
    Jon Aaronson (Tel Aviv University)
    Title: On multiple recurrence and other properties of "nice" infinite measure preserving transformations
  • February 12, 2013
    Sofia Trejo (University of Warwick)
    Title: How to construct complex box mappings

    Abstract: I will talk about the construction of complex box mappings for real analytic
    interval maps. More specifically, I will show that given a real analytic map and a point
    in its postcritical set it is possible to construct complex box mappings, associated to
    the real first return maps, with complex bounds for arbitrarily small scales.

    In the case the map is a non-renormalizable polynomial (not necessarily real) with only
    hyperbolic periodic points, the complex-box mapping can be constructed using the
    Yoccoz puzzle. For real analytic maps we can not guarantee the existence of a Yoccoz
    puzzle. For this reason the construction of the box mapping and the proof of complex
    bounds in this case requires more work.

    Complex bounds are fundamental for the proofs of quasiconformal rigidity, renormalization
    results and ergodic properties.

  • February 19, 2013
    Shirali Kadyrov (University of Bristol)
    Title: Semicontinuity of entropy in homogeneous spaces, escape of mass and Hausdorff dimension.
    Abstract: We discuss, under the action of a diagonal element, the relation between semicontinuity
    of entropy and escape of mass in non-compact homogeneous spaces of rank one. In a noncompact
    case, a limit of a sequence of probability invariant measures may be the zero measure. However,
    if the entropies of the measures in this sequence are uniformly greater than the half of the maximal
    entropy then we show that the limit measure cannot be the zero measure. Moreover, we provide
    bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the
    set of orbits under diagonal action which miss a fixed open set is not full. This is a joint work with
    M. Einsiedler and A. Pohl.
  • February 26, 2013
    Sergey Dobrokhotov (Moscow Institute of Physics and Technology)
    Title: Librations, normal forms and tunneling in quantum double well with magnetic field
    Abstract: A problem for a Schrödinger-particle placed into a symmetric one-dimensional double-well
    potential is considered in many textbooks on Quantum mechanics. It is well known that the level
    splitting in some cases can be computed semiclassically taken into account the presence of the
    small parameter h. A multi-dimensional analogue of the problem has been intensively studied by
    V. Maslov, A. Poljakov, E.M. Harrel, B. Helfer, J. Sjostrand, B. Simon etc. for lowest energy levels.
    The main result of this study is so-called splitting formula for energy E = Aexp(−Jh) with the phase
    J based on a certain classical trajectories known as instanton. The constructive but complicated
    formula of the amplitude A also connected with the instanton was given in the papers by
    S. Yu. Dobrokhotov, V. Kolokoltsov and V. Maslov. In this talk we show that the final splitting
    formula takes more natural and simple form if one changes the instanton by so-called libration
    (unstable closed trajectories) and use the normal forms coming from classical mechanics
    (J. Bruning, S. Yu. Dobrokhotov, E. S. Semenov, Unstable closed trajectories, librations and
    splitting of the lowest eigenvalues in quantum double well problem, Regular and Chaotic
    Dynamics, 11, 2, (2006), 167-180). Also we discuss the following (non trivial) question: what happens
    with the splitting in the presence of magnetic field and consider the application to a problem about
    propagation of wave packets in nanowires.
    This work was done together with J.Brüning and R.V.Nekrasov.

  • March 5, 2013
    Dalia Terhesiu (University of Roma Tor vergata)
    Title: Mixing for some non-uniformly hyperbolic systems with infinite measure
    Abstract: Abstract: In work in progress with Carlangelo Liverani, we obtain mixing and
    mixing rates for the class of non-uniformly hyperbolic systems with a countable Markov
    partition, preserving an infinite measure. Our method consists of a combination of the
    framework of operator renewal theory (as introduced in the context of dynamical systems
    by Sarig) with the framework of function spaces of distributions (developed in the recent
    years along the lines of Blank, Keller and Liverani). In this sense, an important part of our
    work is devoted to establishing the spectral gap for the induced map of the original system.
    Although, at present we only have results for the Markov case, we are confident that the
    method can be generalised to other important classes of transformations. In this talk, I will
    present the main results and describe the main steps of the construction.
  • March 12, 2013
    Anthony Dooley (University of Bath)
    Title: CPE actions of amenable groups
    Abstract: It is well known that Bernoulli actions (of $\mathbb Z$) of the same entropy are
    metrically isomorphic, but if one abandons the hypothesis that the actions are Bernoulli,
    Ornstein showed that one can have uncountably many non-isomorphic actions, each with
    completely positive entropy, of any given entropy. I shall discuss work with Golodets,
    where we study CPE actions of amenable groups; there is a version of Ornstein's result
    for discrete amenable groups, and we have recently studied CPE actions of nilpotent Lie
    groups in terms of actions of their lattice subgroups. This is also related to Avni's recent
    work on the Thouvenot conjecture.

Term 1

  • October 9, 2012
    Sinisa Slijepcevic (University of Zagreb)
    Title: Universal properties of formally gradient systems

    Abstract: (Joint work with Th. Gallay, Grenoble) Many important dissipative PDE's have a
    gradient structure, which is however on large domains only formal (e.g. the energy fun-
    ctional diverges). We identify some universal properties of that gradient structure, which
    apply to several examples (E.g. the reaction-diffusion equation, the complex Ginzburg-
    Landau equation, the Navier Stokes equation). By using that, we describe the invariant
    measures, relaxation times, etc., and show that in spatial dimensions d=1,2, such systems
    behave like gradient ones in a statistical sense, and find counterexamples in d>=3. As an
    example, we study implications to the unforced Navier Stokes equation on a cylinder.
  • October 16, 2012
    Richard Sharp (University of Warwick)
    Title: Noncommutative geometry and measures on limit sets

    Abstract: Noncommutative geometry aims describe a wide range of mathematical objects
    in terms of C*-algebras. We will discuss how this applies to measures on limit sets, including
    Patterson-Sullivan measures on certain Kleinian groups.
  • October 23, 2012
    Andrew Barwell (University of Bristol)
    Title: On Shadowing and Limit Sets in Quadratic Maps

    Abstract: Shadowing is an important property in dynamical systems, as it tells us that a
    system is stable under small perturbations of its orbits. Related to shadowing is the
    notion of an internally chain transitive set: a set in which every pair of points can be
    linked either by the orbit of a point, or by a perturbed orbit; a seemingly unrelated
    concept is that of an omega-limit-set - the attracting set of the orbit of a point in
    the space.

    Using symbolic dynamics, we will show that a quadratic map, whose Julia sets is a
    dendrite, has shadowing. Furthermore, for these maps the omega-limit sets are
    precisely the internally chain transitive sets. It turns out that this is not a coincidence.

  • October 30, 2012
    Lara El Sabbagh (University of Warwick)
    Title: A Lambda-lemma for Normally Hyperbolic Manifolds and an Application to Diffusion

    Abstract: Roughly speaking, a lambda-lemma for normally hyperbolic manifolds asserts that, given a
    smooth manifold M and a diffeomorphism f: M ---> M and a normally hyperbolic submanifold N of M,
    if Gamma is a submanifold that transversely intersects the stable manifold of N, then, under iteration
    by f, Gamma ''approaches" the foliation of the unstable manifold of N in a suitable topology.
    We will present a lambda-lemma for normally hyperbolic manifolds, and deduce a result on diffusion
    from it: we prove an existence result on drifting orbits along a transition chain of invariant dynamically
    minimal sets contained in a normally hyperbolic manifold. This is without any assumption on the
    geometrical nature of the invariant sets (in particular, they do not need to be submanifolds).
    As a particular case, we recover the well-known example of Arnold.
  • November 6, 2012
    Kelly Yancey (University of Illinois at Urbana-Champaign)
    Title: Generic Homeomorphisms

    Abstract: We will start by discussing the history of generic results in the setting of measure-preserving
    automorphisms in ergodic theory. Then we will specifically discuss weakly mixing homeomorphisms that
    are uniformly rigid and give generic type results for this class of homeomorphisms defined on the two
    torus and the Klein bottle.
  • November 13, 2012
    Vadim Kaimanovich (University of Ottawa)
    Title: Continuity of the asymptotic entropy

    Abstract: The talk is based on a joint work with Anna Erschler. We prove continuity of the asymptotic

    entropy for a large class of random walks on groups with hyperbolic properties.

  • November 20, 2012
    Zeng Lian (Loughborough University)
    Title: Lyapunov exponents and Multiplicative Ergodic Theorem for random systems in a separable Banach space.

    Abstract: Lyapunov exponents play an important role in the study of the behavior of dynamical
    systems, which measure the average rate of separation of orbits starting from nearby initial points.
    They are used to describe the local stability of orbits and chaotic behavior of systems.
    Multiplicative Ergodic Theorem provides the theoretical fundation of Lyapunov exponents, which
    gives the fundamental information of Lyapunov Exponents and their associates invariant subspaces.
    In this talk, I will report the work on Multiplicative Ergodic Theorem (with Kening Lu), which is
    applicable to infinite dimensional random dynamical systems in a separable Banach space.
    The system could be generated by, for example, random partial differential equations.
  • November 27, 2012
    Philip Boyland (University of Florida)
    Title: Rotation sets for beta shifts and torus homeomorphisms
    Abstract: (PDF Document)
  • December 4, 2012 (Cancelled)
    Andrzej Bis (University of Warwick)