Description: The standard format is a talk in two halves separated by a break, with the first half aimed at nonspecialists.
Organizers: Peter NandoriJoe Auslander
When: Thursdays @ 2:00pm - 4:00pm
Where:  Math 1311
Dynamics Website:
Pre-2012 Archives:

Archives: 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017

  • Infinite mixing for one-dimensional maps with an indifferent fixed point

    Speaker: Marco Lenci (Universita' di Bologna) -

    When: Thu, August 31, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: In the first part of the talk, I will give some background on the
    question of mixing for dynamical systems preserving an infinite
    measure (a.k.a. 'infinite mixing'). Then I will recall and discuss the
    definitions of 'infinite-volume mixing' that I have introduced in
    recent years, with a survey on some examples of dynamical systems
    which verify or do not verify such definitions. Among these examples
    there will be one-dimensional intermittent maps, the subject of recent
    work with C. Bonanno and P. Giulietti.
    In the second part of the talk, I will better state the results for
    the intermittent maps: they comprise a class of expanding maps of
    [0,1] with a 'strongly neutral' fixed point in 0 and a class of
    expanding maps of the real line with strongly neutral fixed point at
    infinity. I will give a sketch of how some of the definitions of
    infinite-volume mixing are proved or disproved. Finally I will show
    how one property, called global-local mixing, entails certain limit
    theorems for our intermittent maps.
  • Ground states for Frenkel-Kontorova models on quasicrystals

    Speaker: Rodrigo Trevino (UMD) -

    When: Thu, September 7, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: The Frenkel-Kontorova model was first proposed in the 1930's to describe the structure and dynamics of a crystal lattice in the vicinity of a dislocation core, and by now has found many uses outside of solid state physics. Viewed from a dynamical systems point of view, it exhibits a lot of rich behavior tied to all sorts of great theories (e.g. KAM theory and Aubry-Mather theory) and fundamental open questions (e.g. Lyapunov exponents for the standard map).

    I will talk about this model in the setting where the crystal is aperiodic. In this setting, most of the dynamics are no longer available, but some tools developed to study the (periodic) classical model are still useful. I will talk about how one of them in particular, the so-called anti-integrable limit, is useful to find ground states (also known as equilibrium configurations). No background on the model will be assumed.
  • Mixing and local limit theorem for some hyperbolic flows

    Speaker: Peter Nandori (UMD) -

    When: Thu, September 28, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: We consider a special flow over a mixing map with some hyperbolicity.
    In case the roof function is square integrable, we find a set of conditions, under which the flow is mixing and also satisfies the local limit theorem. In case the roof function is non-integrable, we identify another set of conditions that imply Krickeberg mixing. The most important condition is the local limit theorem for the underlying map. We check that the conditions are satisfied for some examples, such as Axiom A flows, Sinai billiards, geometric Lorenz attractors (finite measure case) and suspensions over Pomeau-Manneville maps (finite and infinite measure cases). The talk is based on joint work with Dmitry Dolgopyat.
  • Fourier decay and spectral gaps on hyperbolic surfaces - UNUSUAL TIME

    Speaker: Semyon Dyatlov (MIT) -

    When: Tue, October 17, 2017 - 2:00pm
    Where: Kirwan Hall 1308

    View Abstract

    Abstract: Let $M$ be a nonelementary convex co-compact hyperbolic surface. It is well-known that the Selberg zeta function $Z_M(s)$ has a zero at $s=\delta$ and no zeroes to the right of it, where $0<\delta<1$ is the dimension of the limit set. I will show that $M$ has a spectral gap of some size $\varepsilon>0$ depending only on $\delta$, that is $Z_M(s)$ has only finitely
    many zeroes in $\{\Re s > \delta-\varepsilon\}$. An application is an asymptotic counting formula for lengths of closed geodesics with remainder of relative size $O(e^{-\varepsilon t})$.

    The key ingredient of the proof is a Fourier decay bound for the Patterson-Sullivan measure on the limit set. This bound relies on the nonlinearity of the transformations generating the corresponding group as well as bounds on exponential sums which follow from the discretized sum-product theorem. The Fourier decay bound implies a fractal uncertainty principle for the limit set, which in turn gives the gap. This talk will include an introduction to transfer operators on Schottky groups, which are used throughout the proofs.

    This talk is based on joint works with Jean Bourgain and Maciej Zworski.
  • Can you hear the shape of a drum and deformational spectral rigidity of convex axis-symmetric planar domains.

    Speaker: Vadim Kaloshin (UMD) -

    When: Thu, October 19, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: M. Kac popularized the question `Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega$ and
    the associated Dirichlet problem
    \Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.
    The set of $\lambda$’s such that this equation has a solution, denoted
    $\mathcal L(\Omega)$ is called the Laplace spectrum of $\Omega$.
    Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

    Consider the billiard problem inside $\Omega$. Call the length spectrum the closure
    of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of
    the wave trace function, generically, the Laplace spectrum determines the length
    spectrum. In the space of strictly convex axis-symmetric domains we shall discuss
    Sarnak's question whether one can deform such a domain without changing
    its spectrum. This is based joint works with J. De Simoi, A. Figalli and Q. Wei.

    Speaker: Alex Blumenthal (UMD) -

    When: Thu, October 26, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: The Chirikov standard map $F_L$ is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Analysis in this direction is notoriously difficult, and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any value of L.

    I will discuss two related results on a more tractable version of this problem. The first is a kind of ‘finite-time mixing estimate, indicating that for large L and on a suitable timescale, the map $F_L$ is strongly mixing. The second pertains to statistical properties of compositions of standard maps with increasing parameter L: when the parameter L increases at a sufficiently fast polynomial rate, we obtain asymptotic decay of correlations estimates, a Strong Law, and a CLT, all for Holder observables.

    Speaker: Jianlu Zhang (UMD) -

    When: Tue, November 7, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: TBA
  • Asymptotic behavior of Markov operator corresponding to non-expanding piecewise linear maps

    Speaker: Fumihiko Nakamura (Hokkaido University ) -

    When: Thu, November 16, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: The non-expanding piecewise linear map, known as the Nagumo-Sato (NS)
    model, is described as $S_{alpha,beta}(x)=\alpha x+\beta ({\rm
    mod} 1)$, where $0<\alpha,\beta<1$.
    The NS model corresponds to a
    special case of Caianiello's model, and it describes the simplified
    dynamics of a single neuron. It is known that the system shows
    periodic behavior of the trajectory for almost every (alpha,
    beta). Moreover, the map has one discontinuous point when
    $\alpha+\beta>1$, and this leads to a complicated structure for a
    periodicity of the NS model. This structure is called a Farey
    structure which relates to regions in which $S_{\alpha,\beta}$ has a
    periodic point. An important feature of the this structure is that
    there exists a region in which $S_{\alpha,\beta}$ has a periodic point
    with period (m+n) between the region with period m and n. [1]

    In this talk, we will explain above properties for the NS model first,
    and next consider a perturbed dynamical system in which noise is
    applied to the NS model, $x_{t+1}=S_{\alpha,\beta}(x_t) +\xi_t
    ({\rm mod}1)$ for $0<\alpha,\beta<1$, where ${\xi_t}$ are
    independent random variables each having same density g satisfying
    ${\rm supp}{g}=[0,\theta]$ with $\theta >0$. We discuss asymptotic
    properties for the Markov operator[2] corresponding to this perturbed
    model. Especially, we focus on the properties of ``asymptotic
    periodicity'' and ``asymptotic stability''. Then I will introduce our
    main result which tells us which the Markov operator corresponding to
    the perturbed NS model has asymptotic periodicity or asymptotic
    stability for almost all (alpha,beta) and theta. [3]

    The topics of this talk are based on following references:
    [1] F.Nakamura, Periodicity of non-expanding piecewise linear maps and
    effects of random noises, Dynamical Systems, 30(2015), 450-467.
    [2] A. Lasota and M. C. Mackey, Chaos, fractals, and noise: stochastic
    aspects of dynamics, Springer Science and Business Media, 2013.
    [3] F. Nakamura, Asymptotic behavior of non-expanding piecewise linear
    maps in the presence of random noise, (accepted to DCDS-B).
  • TBA

    Speaker: Thibaut Castan (UMD) -

    When: Thu, December 7, 2017 - 2:00pm
    Where: Kirwan Hall 1311

    View Abstract

    Abstract: TBA
  • TBA

    Speaker: (Francoise Pene) -

    When: Thu, February 22, 2018 - 2:00pm
    Where: Kirwan Hall 1311