Dynamical Systems and PDE:s
2010 spring
Seminar program
The seminars will be held mostly on Tuesdays and Thursdays during the semester.
Thursday, June 10, 2010:
14.00-15.00 Maria Saprykina, KTH, Stockholm
Examples of Hamiltonian systems with Arnold diffusion
Abstract:
I shall speak about two new results obtained in collaboration with
Vadim Kaloshin and Mark Levi.
Here is a heuristic description of the problem setting in
``physical terms''. Imagine $n\geq 5$ mathematical penduli attached to
the wall in a line, and rotating. If they are disjoint, the energy
of each pendulum is preserved for all time. Now we join each pair of
neighboring penduli by a thin rubber band. Of course, the total energy
of the system is still preserved. But what happens with the energy of
each individual pendulum? KAM theorem asserts that under some generic
assumptions, for ``most'' initial conditions the energy of each
pendulum will stay close to the initial one for all time.
One of our results asserts that one can find such initial conditions
and such moments of time $t_j$, that at time $t_j$
the $j$-th pendulum moves with almost the total energy of the system.
This is a work in progress.
I shall mostly speak about the following result.
For any $r$ we construct a $C^\infty$--Hamiltonian $H(I,\theta)$,
$I\in \mathbb R3$, $\theta \in mathbb T3$,
which is $C^r$-close to $H_0(I)=\dfrac{\langle I, I \rangle}{2}$ and
has a trajectory dense in a set of maximal Hausdorff
dimension on the unit energy surface.
15.30-16.30 Boris Hasselblatt, Tufts University and European Doctoral Gollege
Legendrian knots and nonalgebraic contact Anosov flows on 3-manifolds
(with Patrick Foulon)
Abstract:
We describe a surgery construction in a neighbourhood of a Legendrian knot
that gives rise to new contact structures preserved by Anosov flows. In
particular, this includes examples on numerous hyperbolic 3-manifolds, and
it gives contact Anosov flows that are not quasigeodesic. As a byproduct,
this also yields quasigeodesic pseudo-Anosov flows.
Wednesday, June 9, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
COURSE: Non-uniformly hyperbolic horseshoes(continuation)
Monday, June 7, 2010:
16.00 - Jean Bourgain, IAS, Princeton
Toral eigenfunctions and distribution of lattice points on spheres
Thursday, June 3, 2010:
14.00-15.00 Michael Björklund, Jerusalem
Multiplicative combinatorics and dynamics
Abstract:
We shall be considering the following two combinatorial problems
of arithmetic nature:
1. Given two large sets in a countable group; how large is the set of
possible products of elements in the two sets?
2. Suppose that the set of possible products is small; does this force
the sets to contain large "regular" sets.
The exact meanings of "large" and "regular" will be defined. We confine
our attention to amenable groups where methods from ergodic theory turn
out to be useful. Even though the results are of a purely combinatorial nature, the proofs rely heavily on techniques from ergodic theory, structure theory of groups and harmonic analysis.
Joint work with A. Fish (Madison).
15.30-16.30 Artur Avila, Universitè Paris 6
KAM, Lyapunov exponents and the spectral dichotomy for one-frequency Schrödinger operators
Abstract:
We explain the proof of the almost reducibility conjecture for
Diophantine frequencies, which characterizes, for one-frequency
SL(2,R) cocycles, the domain of applicability of KAM methods in terms
of the Lyapunov exponent of complexifications.
Wednesday, June 2, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
COURSE: Non-uniformly hyperbolic horseshoes(continuation)
Tuesday, June 1, 2010:
14.00-15.00 Yakov Pesin, Penn State University
Stable Ergodicity of Partially Hyperbolic Diffeomorphisms: The
Dissipative Case.
Abstract:
I describe two "competing" methods to show that a given
partially hyperbolic diffeomorphism is stably ergodic (i.e., it is
ergodic along with any of its sufficiently small perturbations). One
of them relates the problem to the global estimates of the action of
the system along its central direction while another one deals with a
more delicate estimates using Lyapunov exponents in the central
direction.
15.30-16.30 Pierre Berger, Universitè Paris 13
Abundance of one dimensional non-uniformly hyperbolic attractors for
surface endomorphisms
Abstract:
We present a (new) proof of the existence of a non uniformly
hyperbolic attractor for a Lebesgue positive set of parameters $a$ in the
family of endomorphisms:
\[(x,y)\mapsto (x2+a+y,0)+B(x,y),\]
where $B$ is any fixed $C2$ small function. For $B=0$, this is the
Jakobson's theorem. For $B=b\cdot (0,x)$, we get the Benedicks-Carleson
(B-C) theorem for the Hénon map.
The proof is done thanks to analytical and probabilistic tools of
(B-C) in the geometric and combinatorial formalism of Yoccoz puzzles
generalized in an algebraic way (pseudo-semi-group). These
theorems are notably generalized to the $C2$-case and to the
endomorphisms case. The theorem is an answer to a question of Pesin-Yurchenko
about reaction-diffusion PDEs regarding applied mathematics.
Friday, May 28, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
COURSE: Non-uniformly hyperbolic horseshoes(continuation)
Thursday, May 27, 2010:
9 .30-10.30 Wei-min Wang, Université Paris Sud 11
Supercritical NLS: quasi-periodic solutions and almost global existence
Abstract:
We construct quasi-periodic solutions and prove almost
global existence for supercritical NLS on the d-torus
(d arbitrary). The main new ingredient is a fine analysis of
the resonance geometry using algebraic equations. Time permitting, we will also discuss critical Sobolev exponents and S_1 isometry.
11:00-12.00 Rafael de la Llave, University of Texas,Austin
Periodic and almost periodic breathers
in Hamiltonian lattice systems.
Abstract:
We consider Hamiltonian systems placed at
points in a lattice. We assume that each of
the systems has a positive measure of
KAM tori which are non degenerate and a
hyperbolic fixed point. We can form
whiskered tori by taking the product of
a KAM torus at one site and hyperbolic
fixed points at the others.
This uncoupled system is subject to
a Hamiltonian perturbation such that the
effect of one site on the others decays
with the distance among them
We show A) For small couplings,
there is a positive measure
of frequencies whose whiskered tori survive.
B) Consider a probability measure among the
tori whiskered tori that survive.
There is a full measure set of
frequencies such that we can find a
multi-center breather with this frequency.
Both theorems are deduced by creating a
set up that allows to deal with infinite dimensional
systems with local interactions and proving
an "a-posteriori" KAM theorem.
We also study other invariant manifolds
associated to the whiskered tori and
their role in the creation of chaotic behaviors.
14.00-15.00 David Damanik, Rice University, Houston
What determines the spreading of a wavepacket?
Abstract:
This question has been asked by numerous physicists and mathematicians. While there are classical approaches to the study of the dynamics of the Schr\"odinger equation, they apply only in certain situations and an understanding of the general principles governing the time-evolution of a wavepacket evolving according to this fundamental equation has yet to be obtained. In this talk we describe some recent developments regarding this issue and address in particular a question of Yoram Last about a connection between the box counting dimension of the spectrum of the underlying Schr\"odinger operator and upper bounds for wavepacket spreading. We give a negative answer to this question and explain some dynamics aspects of its proof.
15.30-16.30 Anders Szepessy, KTH, Stockholm
How accurate is molecular dynamics?
I will show that Ehrenfest and Born-Oppenheimer
molecular dynamics accurately approximate
observables based on the
time-independent Schrödinger equation,
in the limit of large ratio of nuclei and electron masses.
The derivation, based on a Hamiltonian system interpretation of the
time-independent Schrödinger equation and stability of the corresponding
Hamilton-Jacobi equation,
bypasses the usual separation
of nuclei and electron wave functions
and gives a different perspective on
Hamiltonian systems in molecular dynamics modeling.
Wednesday, May 26, 2010:
9 .30-10.30
Yakov Pesin, Penn State University
Thermodynamics of Towers of Hyperbolic Type
Abstract: I introduce a class of continuous maps of compact topological spaces admitting inducing schemes of hyperbolic type and discuss thermodynamic formalism associated with such schemes, i.e., describe a class of real-valued potential functions, which possess a unique equilibrium measure minimizing the free energy. The results apply to certain multidimensional maps allowing Young's tower constructions.
11:00-12.00 Benoit Grebert, Universitè de Nantes
KAM for the semilinear quantum harmonic oscillators
Abstract:
We prove an abstract KAM theorem for infinite dimensional Hamiltonians systems
which extends previous works by S. Kuksin and by J. Pöschel. As an
application we show that some 1D nonlinear Schrödinger
equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.
14.00-15.00 Artur Avila, Universitè Paris 6
Absence of critical energies for typical one-frequency Schrödinger operators
Abstract: We explain the parameter exclusion process leading to the title, focusing on the transversality estimate (a non-positivity result for the derivative of the Lyapunov exponent).
15.30-16.30 Thomas Kappeler, Universität Zürich
On the asymptotics of the Birkhoff map
In this talk I will outline the proof of
a conjecture of Kuksin and Perelman saying
that the Birkhoff map of the Korteweg-deVries
equation on the circle, which transforms
this equation globally into normal form,
is a 1-smoothing perturbation of the (weighted)
Fourier transform. This property allows
to study random perturbations with damping
of the KdV equation in normal coordinates.
Tuesday, May 25, 2010:
9 .30-10.30
Walter Craig, McMaster University, Hamilton
Lagrangian invariant tori for Hamiltonian systems with
infinitely many degrees of freedom
Abstract:
This talk describes a construction of invariant tori of full dimension
for a version of the nonlinear Schroedinger equation posed on a
lattice, in the form
-i\dot q_n = \mu_n q_n + |q_n|^2 q_n + \epsilon (q_{n+1} + q_{n-1}) ,
n \in Z_+
where \mu_n = n and where q_0 = 0 is the boundary condition at n = 0.
This problem is an infinite-dimensional Hamiltonian dynamical system,
and the proof uses an extension of the approach of classical KAM theory,
which is augmented by a higher order normal form controlling the
curvature of the action-frequency map, and a multiple scale frequency
analysis. This normal form is available under higher order Melnikov-like
nonresonance conditions. A principal point of our result
is that no external parameters are employed in order to satisfy
the infinite-dimensional nonresonance conditions, instead this is
performed naturally through the action - frequency map. The
construction is a general procedure, which extends to a broad
class of nonlinear lattice equations and other Hamiltonian
dynamical systems possessing infinitely many degrees of freedom.
This is work in collaboration with Jiansheng Geng.
11:00-12.00 Alexander Bufetov, Institute of Mathematics, Moscow, and Rice University, Houston
Limit theorems for translation flows
Abstract:
Consider a compact oriented surface of genus at least two
endowed with a holomorphic one-form.
The real and the imaginary parts of the one-form define two foliations on the
surface, and each
foliation defines an area-preserving translation flow. By a Theorem of H.Masur
and W.Veech,
for a generic surface these flows are ergodic. The talk will be devoted to
the speed of convergence in the ergodic theorem for translation flows.
The main result, which extends earlier work of A.Zorich and G.Forni,
is a multiplicative asymptotic expansion for time averages of Lipschitz
functions.
The argument, close in spirit to that of G.Forni, proceeds by
approximation of ergodic integrals by special holonomy-invariant
Hoelder cocycles on trajectories of the flows.
Generically, the dimension of the space of holonomy-invariant
Hoelder cocycles is equal to the genus of the surface, and the ergodic integral
of a Lipschitz function can be approximated by such a cocycle up to terms
growing slower than any power of the time.
The renormalization effectuated by the Teichmueller geodesic flow
on the space of holonomy-invariant Hoelder cocycles allows one also to obtain
limit theorems
for translation flows: it is proved that along certain
sequences of times ergodic integrals, normalized to have variance one, converge
in distribution to a non-degenerate compactly supported measure.
The argument uses a symbolic representation of translation flows as suspension flows over Vershik's automorphisms, a construction similar to one proposed by S.Ito.
14.00-15.00 Margaret Beck, University of Boston
Time-periodic parabolic PDEs on unbounded spatial domains: a
method for understanding the linear non-autonomous operator and the full
nonlinear dynamics
Abstract:
Floquet theory provides a good way to understand time-periodic
ODEs, and some of that theory holds in the case of time-perioidc parabolic
PDEs on bounded spatial domains, as well. Here we describe related methods
for understanding such equations on unbounded spatial domains, in the
context of systems of viscous conservation laws. In that case, additional
difficulties arise due to the presence of continuous spectrum and the lack
of compactness of the resolvent operators. We indicate how such methods
can be used to prove that time-periodic viscous shocks are nonlinearly
stable.
15.30-16.30 Håkan Eliasson, Universitè Paris 7
KAM for the KP equation
We discuss the perturbation theory for quasi-periodic solutions of the Kadomtsev-Petviashvili equation and the linear Kadomtsev-Petviashvili equation. This is a joint work with Carlos Matheus.
Thursday, May 20, 2010:
14.00-15.00 Jean-Paul Thouvenot, CNRS and UPMC, Paris
Another proof of the norm convergence of nonconventional ergodic averages for commuting transformations
Abstract:
In the ergodic theoretical proof by Furstenberg (1975) of the Szeméredi's theorem on arithmetic progressions, the positivity of the lim inf of "nonconventional ergodic averages" involving powers of a single transformation played a fundamental role. It was not until 2005 that Host and Kra showed that these averages were actually converging. In 2008, T.Tao, and then (very differently) T. Austin proved that the Host-Kra convergence result extended to commuting transformations. We are going to present yet another proof of this last fact using the tool of joinings.
15.30-16.30 Sergei Kuksin, CNRS and Ecole Polytechnique, Paris
Perturbed KdV: results and open problems
Abstract:
For the KdV equation under periodic boundary conditions, perturbed by
Hamiltonian or dissipative perturbations, I will discuss known
results on qualitative behaviour of solutions on long and infinite
time-intervals. I will also mention some open problems which appear
for me the most important.
Wednesday, May 19, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
COURSE: Non-uniformly hyperbolic horseshoes(continuation)
Friday, May 14, 2010:
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katok's Course: KAM and rigidity
Lecture 6:
Setting up of the KAM scheme for the parametric local differentiable rigidity of the
unipotent action of R2 on SL(2,R)\times SL(2,R)/Gamma.
Application of cocycle rigidity. and tame splitting.
Thursday, May 13, 2010:
14.00-15.00 Bassam Fayad, Université Paris 13
Transitive dynamics in the solid torus
Abstract:In a joint work with H. Eliasson and R. Krikorian, we construct
smooth conservative transitive maps in the solid torus T2 x [0,1] with arbitrary (in particular Diophantine) translation frequencies on the boundaries
15.30-16.30 Nicolas Gourmelon, Université de Bordeaux
"Pathwise" perturbations of linear cocycles, and new examples of wild dynamics.
Abstract:
We introduce new perturbative tools inside homoclinic classes. One is a generalization of a famous Franks Lemma, and is based on paths of cocycles. It allows to prescribe derivatives along periodic orbits while controlling the flags of stable and unstable manifolds, depending on the homotopy characteristics of the paths.
Another result by Bonatti and Bochi, gives a full description of the possible path-perturbations of cocycles. From these two results, Bonatti, Crovisier, Diaz and I built examples of self-replicating homoclinic classes.
Wednesday, May 12, 2010:
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katok's Course: KAM and rigidity
Lecture 5:
Cocycle rigidity and tame splitting for the unipotent action of R2 on SL(2,R)\times SL(2,R)/Gamma
Tuesday, May 11, 2010:
14.00-15.00 Dario Bambusi, Universitá Statale di Milano
On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential. (Joint work with Scipio Cuccagna(Modena))
Abstract:
We study small amplitude solutions of nonlinear Klein
Gordon equations with a potential. Under suitable smoothness and
decay assumptions on the potential and a genericity assumption on
the nonlinearity, we prove that all small energy solutions are
asymptotically free. The proof is based on a
combination of Birkhoff normal form techniques and dispersive
estimates.
15.30-16.30 Xiaoping Yuan, Fudan University, Shanghai
KAM theory for unbounded perturbation with application to PDEs.
Abstract:
By establishing a new estimate of the solutions for a
small-divisor equation with large variable coefficients, we prove a
reduction theorem and KAM theorem, and apply them to the study of
spectrum of quantum Duffing oscillator and KAM tori of deriative
nonlinear Schrödinger equation, respectively.
Monday, May 10, 2010:
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katok's Course: KAM and rigidity
Lecture 4: Unitary representation of SL(2,R) and solution of
cohomological equations for the horocycle flow.
Friday, May 7, 2010:
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katok's Course: KAM and rigidity
Lecture 3:
Realization of the KAM scheme for local differentiable rigidity for actions by commuting
partially hyperbolic automorphisms of the torus.
Thursday, May 6, 2010:
14.00 - 15.00 Michael Yampolsky, University of Toronto
Parabolic renormalization
Wednesday, May 5, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
COURSE: Non-uniformly hyperbolic horseshoes(continuation)
Monday, May 3, 2010:
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katok's Course: KAM and rigidity
Lecture 2:
Local differentiable rigidity for actions by commuting automorphisms of the torus:
1. Hyperbolic case. A priori regularity of structural stability.
2. Partially hyperbolic case: setting up the KAM scheme.
Thursday, April 29, 2010:
14.00-15.00 Nalini Anantharaman, Orsay
Semiclassical measures for the Schrödinger equation II : the case of negative curvature.
Abstract:
I will continue my review of the regularity properties for semiclassical measures associated with
solutions of the Schr"odinger equation. This time I will focus on the case of negative curvature. I
will describe work in progress with Gabriel Riviere. One of the results uses Kifer's large deviation
result for trajectories of the geodesic flow, to estimate the proportion of coherent states that do
not become equidistributed in large time. The other result is about the entropy of semiclassical
measures, and has applications in control theory.
15.30-16.30 Kostya Khanin, University of Toronto
Nonrigidity for circle maps with breaks
Tuesday, April 27, 2010:
14.00-15.00 Anatole Katok, Penn State University
Absolutely continuous invariant measures for actions of
higher-rank abelian groups
15.30-16.30 Kristian Bjerklöv, KTH, Stockholm
Quasi-periodic perturbations of some unimodal maps.
Abstract:
We shall present some results on quasi-periodic perturbations
of certain unimodal maps exhibiting an attracting 3-cycle.
Monday, April 26, 2010:
10.15- Michael Benedicks, KTH, Stockholms
Moser's theorem on commuting circle diffeomorphisms.
Two introductory lectures to the course of A. Katok.
14.15 - 16.00 Anatole Katok, Penn State University
Anatole Katoks Course: KAM and rigidity
Lecture 1:
Survey of rigidity properties for smooth actions of abelian groups: hyperbolic, partially
hyperbolic, elliptic and parabolic. Methods used for establishing various flavors of rigidity.
Cocycles, cohomological equations and time changes. General scheme of the KAM method for actions of
higher-rank abelian groups. Overview of the course.
Friday, April 23, 2010:
10.15- Michael Benedicks, KTH, Stockholms
Moser's theorem on commuting circle diffeomorphisms.
Two introductory lectures to the course of A. Katok.
Thursday, April 22, 2010:
NOTE: Change in the seminar
14.00-15.00 Sylvain Crovisier, Université Paris 13, Villetaneuse
Strong homoclinic intersections inside hyperbolic attractors
Abstract: We discuss the geometry of any hyperbolic attractor
having a one-codimensional strong stable bundle.
If it is not contained in a normally contracted
submanifold, one can build by C^(1+a) perturbations
an intersection between the strong stable and the unstable manifolds
of a periodic orbit of the attractor.
This is a joint work with Enrique Pujals.
15.30-16.30 Raphael Krikorian, Université Paris 6
Cocycles not homotopic to the identity
This is a joint work with A. Avila.
LECTURE SERIES
Wednesday, April 21, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collège de France, Paris
Non-uniformly hyperbolic horseshoes.(Continuation of J.-C. Yoccoz' course)
Tuesday, April 20, 2010:
14.00-15.00 Corinna Ulcigrai, University of Bristol
A Gauss map for symbolic sequences in the regular octagon
Abstract:
It is well known that the Farey map and the Gauss map are
related to symbolic codings of the geodesic flow on the modular surface.
Sturmian sequences, which
are symbolic sequences that code the orbit of a linear trajectory in a
square
torus, can be characterized using substitutions generated by the Farey
map. In a
joint work with John Smillie, we considered symbolic sequences which
code a
linear trajectory on the translation surface associated to the regular
octagon
(and more in general regular 2n-gons). We gave a characterization of the
symbolic sequences in terms of "substitutions" generated by the
dynamics of a
map which plays the role of the Farey map in this contest. In further
joint
work, we construct the analogue of the Gauss map and compute the
absolutely
continuous invariant measures. We also give a geometric interpreation
of these
maps as coding of the Teichmueller geodesic flow on an orbifold that
plays the role of the modular surface.
15.30-16.30 Lorenzo Diaz, PUC, Rio de Janeiro
C^1-robust cycles
Abstract:
We will discuss the generation of C^1-robust homoclinic
tangencies and heterodimensional
cycles. We will present some mechanisms for the generation of such
cycles. A key ingredient in these
constructios is the notion of a "blender". We will explain this
concept and its occurrence in dynamics.
Join works with Bonatti and Bonatti-Kiriki.
Thursday, April 15, 2010:
14.00-15.00 Carlos Matheus Santos, Collège de France, Paris
Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models
Abstract: A long-standing conjecture (going back to Smale's work) says that
Axiom A is C1 dense among surface diffeomorphisms. In this talk, we discuss
this conjecture in the context of an interesting family of toy model
examples introduced by M. Benedicks and L. Carleson in their study of Henon
maps.
More precisely, we'll show that Benedicks-Carleson toy models exhibit
"Newhouse phenomena" in the C2 topology, but nevertheless Axiom A is C1
dense among them. To do so, we'll exploit a recent theorem of Moreira
showing that the main known obstruction to hyperbolicity of surface
diffeomorphisms doesn't exist in the C1-topology, namely, there is no C1
stable intersection of Cantor sets.
These results are a joint work with C. Moreira and E. Pujals.
15.30-16.30 Nalini Anantharaman, Ecole Polytechnique, Palaiseau
Semiclassical measures for the Schrödinger equation on the torus.
Abstract:
We study the regularizing properties of the Schrödinger equation associated to the Laplacian on a flat torus. We consider a sequence of initial conditions $u_n$, normalized in L2, and the sequence of probability measures
on the torus, $$\int_01 |e^{it\Delta}u_n(x)|^2 dt) dx.$$
We show that any weak limit is an absolutely continuous measure. This
generalizes a result of Bourgain about eigenfunctions on the torus. Joint work with Fabricio Macia.
LECTURE SERIES
Wednesday, April 14, 2010:
10.15 - 12.00 Jean-Christophe Yoccoz, Collè de France, Paris
Non-uniformly hyperbolic horseshoes
Abstract:
The lecture series will be based on a joint work with J. Palis, recently published in Publ. Math. IHES. The setting is that of homoclinic bifurcations for surface diffeomorphisms, beyond the case considered 25 years ago by Palis and Takens.
For most parameters after the bifurcation, the maximal invariant set under consideration is a non-uniformly hyperbolic saddle-like object, exhibiting dynamical features related to those of Henon attractors.
I will explain in a detailed way the new ideas and tools indroduced to study these non-uniformly hyperbolic horseshoes, which hopefully could be useful in other settings where non-uniformly hyperbolic dynamics seem prevalent.
Tuesday, April 13, 2010:
14.00-15.00 Sylvain Crovisier, Université Paris 13, Villetaneuse
Partial hyperbolicity far from homoclinic tangencies
15.30-16.30 Ian Melbourne, University of Surrey, Guildford
Convergence of fast-slow ODEs to stochastic differential
equations
Abstract: A project started recently with Andrew Stuart investigates the
convergence of certain deterministic systems to a stochastic differential equation.
For (presently over-simplified) fast-slow systems, we prove, under very mild conditions on the fast variables, that the slow-variable solutions
converge to solutions of a stochastic differential equation.
A major difference between our approach and related projects is that we
do not rely on decay of correlations for the fast variables (decay of
correlations for flows is a notoriously difficult and poorly understood
problem). Instead we use invariance principles and large
deviation estimates which have been derived for a very large class of
systems in collaboration with Matthew Nicol.
Friday, April 9, 2010:
9.30-11.00 Jean-Christophe Yoccoz, Collège de France, Paris
Non-uniformly hyperbolic horseshoes
Abstract:
The talk will be an introduction to a lecture series
13.00 - Fredrik Johansson Viklund
Random Loewner Chains(Thesis defense)
Place: KTH, Room F3. Opponent: Ilia Binder, Toronto
Thursday, April 8, 2010:
9.30-10.30 Lorenzo J. Diaz, PUC, Rio de Janeiro
Porcupine-like transitive sets
Abstract:
We study a topologically transitive fibred system over a horseshoe map
that is derived from a homoclinic class. In particular,
this class contains saddles of different indices and hence is not
hyperbolic. It possesses a very rich fibre structure (uncountably many
trivial and uncountably many non-trivial spines). Moreover, we observe
that the spectrum of the central Lyapunov exponents contains a gap. We
will discuss property. Finally, this system is naturally associated
to an iterated function system that is genuinly non-contracting. This
is a joint work with K. Gelfert (UFRJ).
11.00-12.00 Henk Bruin, University of Surrey, Guildford
Dynamics of some interval translation maps
Abstract: Boshernitzan and Kornfeld introduced interval translation
maps (ITM) in 1995, showing that unless the ITM reduces to an interval
exchange transformation (IET), the ITM has a Cantor set as limit
set. Following on from work with Troubetzkoy, on the same three-branch
ITM, I will discuss some results on Hausdorff dimension of this Cantor
set, (non)unique ergodicity and how this is reflected in parameter
space.
14.00-15.00 Giovanni Forni, University of Maryland, College Park
The Kontsevich-Zorich exponents beyond the canonical
measures
Abstract:
We discuss a condition for the non-uniform hyperbolicity
of the Kontsevich-Zorich cocycle with respect to a general
SL(2,R) invariant probability measure and a class of examples where non-uniform hyperbolicity can fail. Part of this work is in
collaboration with C. Matheus and A. Zorich.
15.30-16.30 Keith Burns, Northwestern University, Evanston
Ergodicity of the Weil-Petersson flow
Abstract:
The Weil-Petersson metric on Teichmueller space is an incomplete Riemannian metric with negative curvature, which is invariant under the action of the mapping class group.
It descends to a metric of finite volume on moduli space. The geodesic
flow on the unit tangent bundle of moduli space is ergodic. This is proven by applying the machinery of
Pesin theory as extended by Katok-Strelcyn. The main difficulty is to
verify the conditions of Katok-Strelcyn that allow the theory to be applied. This is joint work with Howie Masur and Amie Wilkinson.
Wednesday, April 7, 2010:
9.30-10.30 Corinna Ulcigrai, University of Bristol
Mixing time-changes of parabolic flows
Abstract:
In this talk we will consider ergodic properties of
parabolic flows and their time reparametrizations. We will mention
mixing properties of (reparametrizations of) linear flows
on surfaces. We will then focus on another fundamental example of
parabolic dynamical systems, Heisenberg nilflows. We prove that there
are time changes which are mxing and that the set of mixing
time-changes is generic in a appropriate sense and can be explicitely
described. The latter is joint work with A. Avila and G. Forni.
11.00-12.00 Stefano Marmi, Scuola Normale Superiore, Pisa
Linearization of generalized Interval Exchange Maps
Abstract: A standard interval exchange map is a one-to-one map of the interval
which is locally a translation except at finitely many
singularities. We define for such maps, in terms of the Rauzy-Veech
continuous fraction algorithm, a diophantine arithmetical condition
called restricted Roth type which is almost surely satisfied in
parameter space. Let T_0 be a standard interval exchange map of
restricted Roth type, and let r be an integer > 2. We prove that,
amongst C^(r+3) deformations of T_0 which are C^(r+3) tangent to T_0
at the singularities, those which are conjugated to T_0 by a C^r
diffeomorphism close to the identity form a C1 submanifold of
codimension (g-1)(2r+1)+s. Here, g is the genus and s is the number of
marked points of the translation surface obtained by suspension of
T0. Both g and s can be computed from the combinatorics of T_0.
Joint work with Pierre Moussa and Jean-Christophe Yoccoz.
14.00-15.00 David Damanik, Rice University, Houston
Dynamical aspects of the spectral anylisis of Schrödinger operators
15.30-16.30 Magnus Aspenberg, Gothenburg
Perturbations of rational Misiurewicz maps
Abstract:
A non-hyperbolic rational map f (from the Riemann sphere to itself) is
called a Misiurewicz map if f has no parabolic periodic points and
such that every critical point c on the Julia set is not contained in
the !-limit set of any critical point. In other words, the last
condion means that critical set on the Julia set is non-recurrent. In
the talk I will present perturbation results for these maps. For
instance, I will outline methods for how to perturb any given
Misiurewicz map f (not being a flexible Lattès map) into a
hyperbolic map. If in addition the Julia set of f is not the whole
Riemann sphere, then f is in fact a Lebesgue density point of
hyperbolic maps. The flexible Lattès maps, which are a special type
of Misiurewicz maps, are harder to perturb because they admit an
invariant line field on the Julia set and therefore more sophisticated
methods have to be used. If time admits I will discuss this and also
present some results (together with J. Graczyk) for semi-hyperbolic
maps introduced by Carleson, Jones and Yoccoz.
Tuesday, April 6, 2010:
9.30-10.30 Raphael Krikorian, Ecole Polytechnique, Paris
On a problem of Michel Herman
Abstract: We prove that an analytic lagrangian torus,
invariant for an analytic hamiltonian system, with diophantine
translation vector is accumulated by other invariant analytic
lagrangian tori. This is a partial answer to a question by Michel
Herman."Large circles on surfaces" Joint work with H. Eliasson and B. Fayad.
11.00-12.00 Mark Pollicott, University of Warwick, Coventry
Large circles on surfaces
Abstract:
It is a classical result that the projection of large
circles in the plane become uniformly distributed with respect to Haar
measure. Similar results hold for surfaces of constant negative
curvature. We consider the problem for some translation surfaces.
14.00-15.00 Carlos Matheus Santos, Collège de France, Paris
Lyapunov spectrum of Kontsevich-Zorich cocycle over SL(2,R)
orbits of square-tiled cyclic covers
Abstract:
In this talk we'll discuss the Lyapunov exponents of
Kontsevich-Zorich (KZ) cocycle over SL(2,R) orbits obtained by cyclic
covers of the Riemann sphere with 4 branch points. In particular,
during the first part of the talk, we'll describe completely KZ
cocycle over two special orbits with totally degenerate spectrum,
while the second part will focus on the fact that these orbits form an
infinite family of examples where the non-negative of Lyapunov
exponents of KZ cocycle are either zero or positive with multiplicity
2 (at least). This shows that the non-typical orbits (wrt Masur-Veech
measure) can have essentially any type of bad behavior, in contrast to
the simplicity and non-uniform hyperbolicity results of Forni and
Avila-Viana.
The first part of the talk is based on joint work with J. C. Yoccoz,
while the second part is based on joint work with G. Forni and
A. Zorich.
15.30-16.30 Alexey Teplinsky, National Academy of Sciences of Ukraine, Kiev
Hyperbolic horseshoe for renormalizations of circle
diffoemorphisms with a break
Abstract:We consider the renormalization operator acting on commuting
pairs, which correspond to circle diffeomorphisms with a break (or
with several breaks lying on the same trajectory). It is known that
its trajectories converge exponentially fast to the invariant 2-d
manifold built of fractional-linear pairs. Inside that manifold, we
show the existence of a hyperbolic structure that possesses a
remarkable symmetry; the whole construction is very visual. This
knowledge allows one to prove a rigidity result for circle
diffeomorphisms with a brake.
Joint work with K. Khanin
Tuesday, March 30, 2010:
14.00-15.00 Viviane Baladi, ENS. Paris
Anisotropic Sobolev spaces adapted to piecewise hyperbolic dynamics
Abstract:
Strong ergodic properties (such as exponential mixing)
have been proved for various smooth dynamical systems by
first obtaining a spectral gap for a suitable
"transfer" operator acting on an appropriate Banach space.
Some natural dynamical systems, such as discrete or continuous-time
billiards, are only piecewise smooth, and the discontinuities pose
serious technical problems. In a recent paper with Sebastien Gouezel,
we show that classical tools such as complex interpolation
on anisotropic Sobolev-Triebel spaces, and an old result of
Strichartz on Fourier multipliers, can solve
those problems, under a bunching condition.
(The bunching condition replaces a smoothness
assumption on the stable bundles which was necessary in
a previous work.)
I will also briefly discuss work in progress with
Balint and Gouezel on the one hand, and Liverani on the
other, the ultimate goal of which is to establish
exponential mixing for continuous-time 2-d Sinai billiards.
15.30-16.30 Mark Pollicott, University of Warwick, Coventry
Factors of measures for shifts
Abstract: This talk considers an interesting problem which occurs in the study of Hidden Markov chains. Namely, to understand what happens to Markov and Gibbs measures when they are factored from one shift space to another by a simple map which collapses states (i.e., one block factor maps). This is joint work with Thomas Kempton.
Thursday, March 25, 2010:
14.00-15.00 Tomas Johnson, Uppsala University
Dynamics of the universal area-preserving map associated with period
doubling I: Hyperbolicity
Abstract: In 1984 Eckmann-Koch-Wittwer gave a computer-assisted proof of the
existence of a universal area-preserving map - a map with orbits of
all binary periods. This universal map is a fixed point of a locally
hyperbolic renormalization operator.
We demonstrate that the universal map admits a ``bi-infinite
heteroclinic tangle'': a sequence of periodic points whose stable and
unstable manifolds intersect transversally. This yields the existence
of oscillatory motions. We also consider maps in some neighbourhood of
the universal map, and show that the third iterate for all maps close
to the universal one admits a horseshoe, and provide bounds on its
Hausdorff dimension.
This is joint work with Denis Gaidashev
15.30-16.30 Denis Gaidashev, Uppsala University
Dynamics of the universal area-preserving map associated with period
doubling II: Stability
Abstract: We study the stable dynamics of infinitely renormalizable maps. For
all such maps we prove the existence of a "stable" invariant Cantor
set such that the Lyapunov exponents are zero, and estimate its
Hausdorff dimension. This stable set is an analogue of the Feigenbaum
attractor for dissipative Henon-like maps.
We also show that there exists a submanifold of finite codimension in
the renormalization local stable manifold, such that for all maps on
this submanifold the stable set is ``weakly rigid'': the dynamics of
any two maps in this submanifold, restricted to the stable set, is
conjugated by a bi-Lipschitz transformation that preserves the
Hausdorff dimension.
This is joint work with Tomas Johnson
Tuesday, March 23, 2010:
14.00-15.00 Marco Martens, Stony Brook University
Henon renormalization II
15.30-16.30 Claire Chavaudret, Université Paris VII
Reducibility and almost reducibility of quasi-periodic cocycless
Abstract: Quasiperiodic cocycles are cocycles over an irrational rotation
on a torus, and they are said to be reducible if they can be
conjugated to a constant. We present some results on this topic.
Thursday, March 18, 2010:
14.00-15.00 Marco Martens, Stony Brook University
Henon renormalization
15.30-16.30 Alexey Teplinsky, National Academy of Sciences of Ukraine, Kiev
Smooth conjugacy of circle diffeomorphisms
Abstract:
The recent developments in the title topic will be discussed. I will present my results (most of them joint with K.Khanin) concerning the Herman theory and the rigidity theory for circle diffeomorphisms with a break or with a critical point.
The common assumption is the low smoothness of the diffeomorphisms, i.e. $C^{2+\alpha}$.
Tuesday, March 16, 2010:
14.00-15.00 Sergey Dobrokhotov, A. Ishlinski Institute for Problems in Mechanics of Russian Academy of Sciences and Moscow Institute of Physics and Technology
Explicit asymptotics of localized solutions to linearhyperbolic systems
Abstract: We suggest new method for the construction asymptotic solutions to Cauchy problems with
localized initial data for the multidimensional linear hyperbolic systems with variable
coefficients, in particular the linearized Shallow water equations. Our main result based
on the generalization of the Maslov canonical operator is explicit formulas which establish
the connection between initial perturbations and wave profiles near the fronts, including
the neighborhood of focal points and vortical solutions. As application we consider tsunami
waves and mesoscale vortices (typhoons and hurricanes) in the atmosphere.
15.30-16.30 Rafael de la Llave, University of Texas, Austin
Invariant objects in coupled map lattices
Abstract:
We consider infinite dimensional systems that consist of copies of
a finite dimensional system at each point in the lattice coupled by
interactions which decrease fast enough. These objects have appeared
in applications under the name of ``coupled map lattices'', ``oscillator
networks'' and in discretizations of PDE's.
We consider in detail hyperbolic systems and their invariant manifolds.
When the system is Hamiltonian, we also consider whiskered invariant tori
and their invariant manifolds. The method allows to consider the persistence
of tori with finitely many or infinitely many frequencies.
Joint work with E. Fontich, P. Martin, Y. Sire (previous work with
M. Jiang)
Thursday, March 11, 2010:
14.00-15.00 Francois Ledrappier, University of Notre Dame
Entropies for covers of compact Riemannian manifolds Part I
15.30-16.30 Francois Ledrappier, University of Notre Dame
Entropies for covers of compact Riemannian manifolds Part II
Tuesday, March 9, 2010:
14.00-15.00 Manfred Denker, Universität Göttingen
Erdös-Renyi Laws for Dynamical Systems
Abstract: An almost immediate consequence of large deviation for Birkhoff sums
is the Erdö-Renyi law which is a type of ergodic theorem for maxima of
partial sums of $\log n$ successive terms where the maximum is taken over starting points $T^j(x)$ for $0\le j\le n-\log n$. I will explain this and related questions and show the result for Gibbs-Markov dynamics. The result is joint work with Z. Kabluchko.
15.30-16.30 Andreas Knauf, Friedrich-Alexander-Universität, Erlangen
Hamiltonian motion in (non-)random potentials
Thursday, March 4, 2010:
14.00-15.00 Margaret Beck, University of Boston
Understanding metastability using invariant manifolds
Abstract:
Metastability refers to transient dynamics that persist for long times. More precisely, suppose a PDE has a globally attracting state, meaning that, for any initial condition, the solution will asymptotically approach that state. It can happen that, on its way to the state, the solution spends a long period of time near another, possibly unstable, state. This happens, for example, in the Navier-Stokes equation in two spatial dimensions and Burgers equation in one spatial dimension, both with small viscosity. I will explain how, in the context of Burgers equation, this behavior can be understood using certain global invariant manifolds in the phase space of the PDE
15.30-16.30 Mike Todd, University of Boston
Transience in dynamical systems
Abstract:The statistical behaviour of a dynamical system $(X,f)$ can be analysed using thermodynamic formalism. This usually involves taking some potential $\phi$ and considering the pressure and equilibrium states of $(X,f, \phi)$. Singular behaviour of $(X,f)$ can show up in the non-smoothness of the pressure w.r.t. $\phi$. This is referred to as a `phase transition'. In the case where $(X,f)$ is a countable Markov shift, phase transitions have been related to the notion of `transience'. In this talk I'll discuss joint work with G. Iommi, where we study transient phenomena in the compact setting.
Tuesday, March 2, 2010:
Double standard maps --- kneading theory and density of hyperbolicity
14.00-14.45 Ana Rodrigues, Universidade do Porto
Background, kneading theory and quasisymmetric conjugacy
15.15-16.00(NOTE the starting time!) Michael Benedicks, KTH, Stockholm
Quasiconformal perturbations and Teichmüller theory. Extensions to k-fold standard maps
Thursday, February 25, 2010:
14.00-15.00 Boris Kruglikov, University of Tromsö
Dynamics of piece-wise affine maps and applications.
Abstract: I recall my old results (joint with M.Rypdal) on entropy bounds for piece-wise affine systems via Lyapunov exponents and multiplicity entropy and discuss some applications in physics and biology.
15.30-16.30 Simon Kristensen, University of Aarhus
Rotations revisited
Abstract:
One of the first ergodic dynamical systems we encounter is
an irrational rotation of the circle. Although these systems have been
intensively studied, they still contain many surprises and open
problems. We will discuss some of these, with a particular emphasis
on certain shrinking target problems
Tuesday, February 23, 2010:
14.00-15.00 Carlangelo Liverani, Universita degli studi di Roma, Tor Vergata
Something I learned from Dolgopyat (he claims is Varadhan's)
Abstract: I will illustrate a general strategy to obtain limit theorems in some simple examples related to random walk
15.30-16.30 Ale Jan Homburg, University of Amsterdam
Forced circle diffeomorphisms
Abstract:We discuss the dynamics of randomly perturbed circle diffeomorphisms and of circle diffeomorphisms that are forced by an expanding circle map
PLEASE NOTE THE REVISED AND EARLIER STARTS OF THE FRIDAY TALKS!
Friday, February 19, 2010:
9.00-10.00 Klaus Sdmidt, University of Vienna
Entropy and periodic points
10.30-11.30 Mark Pollicott, University of Warwick, Coventry
Asymptotic escape rates for subshifts
13.00-14.00 Anatole Katok, Penn State University, University Park
From Pesin theory to measure rigidity to Zimmer's program
14.15-15.15 Carlangelo Liverani, Universita degli studi di Roma, Tor Vergata
Concerning the derivation of the Fourier Law
Thursday, February 18, 2010:
9.30-10.30 Henk Bruin, University of Surrey, Guildford
Monotonicity of entropy for families of polynomials on the interval
11.00-12.00 Giovanni Forni, University of Maryland, College Park
Mixing for reparametrizations of Heisenberg nilflows
13.30-14.30 Feliks Przytycki, IMPAN, Warsaw
Analyticity of pressure for rational maps and maps of the interval
15.00-16.00 Ana Rodrigues, IUPUI, Indianapolis and Universidade do Porto
Simple conjugacy invariants for braids
16.15-17.15 Anders Öberg, Uppsala University
Uniqueness, mixing and Bernoullicity of g-measures
Wednesday, February 17, 2010:
9.30-10.30 Viviane Baladi, ENS, Paris
Linear response for generic nonuniformly hyperbolic unimodal maps
11.00-12.00 Andreas Knauf, Friedrich-Alexander-Universität,Erlangen
Topological and geometric ideas in scattering theory
13.30-14.30 Matthew Nicol, University of Houston
Borel-Cantelli lemmas for nonuniformly expanding maps
15.00-16.00 Jeff Steif, Chalmers, Gothenburg
The very many different faces of the T T-inverse process
16.15-17.15 Francois Ledrappier, University of Notre Dame, Indiana
Entropies of covers for compact manifolds
Tuesday, February 16, 2010:
9.30-10.30 José Alves, Universidade do Porto
From mixing rates to recurrence times
11.00-12.00 Jairo Bochi, PUC, Rio de Janeiro
Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms
13.30-14.30 Manfred Einsiedler, ETH, Zürich
An ergodic proof of Duke's theorem
Abstract:
Duke proved equidistribution of certain packets of periodic geodesic
orbits on the modular surface using subconvexity of L-functions. We
will outline an ergodic theoretic proof (with a minimal number
theoretic input) which is closely related to the method of Linnik and
Skubenko from around 1950. This is joint work with E.Lindenstrauss,
Ph.Michel, and A.Venkatesh.
15.00-16.00 Anders Karlsson, KTH, Stockholm and University of Geneva
Noncommutative ergodic theorems
Thursday, February 11, 2010:
14.00-15.00 Michael Melgaard, Dublin Institute of Technology/Uppsala University
Recent results on models of Quantum Chemistry, I
Abstract: An introduction to models of Quantum Chemistry is given
and recent results are discussed, in particular for systems with external fields.
15.30-16.30 Mattias Enstedt, Uppsala University
Recent results on models of Quantum Chemistry, II
Abstract:Recent developments concerning solutions to the Hartree-Fock equations
for N-electron Coulomb systems with quasirelativistic kinetic energy are discussed.
We establish existence of a ground state and excited states when the total charge
of $K$ nuclei is greater than $N-1$ and smaller than a critical charge.
Tuesday, February 9, 2010:
14.00-15.00 Reiner Lauterbach, Universität Hamburg
Some open problems in equivariant dynamics.
Abstract: In this talk we will give a short introduction into the theory of equivariant bifurcation and equivariant dynamics.
We will focus on some questions which are important for the theory. In applications these points may not be in the center of the interest.
We will dicuss some recent results and several open questions.
15.30-16.30 Hiroki Takahasi, Kyoto University
Prevalent dynamics at the first bifurcation of the Henon family
Abstract:
We investigate the dynamics of strongly dissipative Henon maps
around the first bifurcation parameter a^* at which uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. Developing
the idea of dynamically critical points of Benedicks and Carleson,
we show that a^* is a full density point of the set of parameters
for which Lebesgue almost every initial point diverges to infinity
under positive iteration.
Thursday, February 4, 2010:
14.00-15.00 Matthew Nicol, University of Houston
Extreme value theory and hitting time statistics for dynamical systems with some degree of hyperbolicity
Abstract:
Suppose $T: X \rightarrow X$ is a map of a metric space preserving a probability measure $\mu$ and $\phi: X\rightarrow R$ is a H\"older observation on $X$. We may define $M_n (x)=\max \{ \phi(x), \phi ( T x), \ldots , \phi (T^n x) \}$, the sequence of successive maxima. Extreme value theory is concerned with the existence and type
of nondegenerate distributions $G(v)$ obtained by scaling $M_n$ by constants $a_n>0,b_n$
in the sense that $\mu (a_n (M_n -b_n)\le v )\rightarrow G(v)$. For iid random variables there are only three possible limiting distributions, Types
I, II and III. This is a similar phenomena to the universality of the central limit theorem. Extreme value theory has implications for hitting and return time statistics by taking $\phi$ to be a function monotonically decreasing as a function of distance from a distinguished point $x_0\in X$, for example
$\phi (x)=-\log d(x,x_0)$. We describe recent results on extreme value theory for certain classes of nonuniformly expanding maps, hyperbolic billards, lozi-type maps and suspension flows. Some of this work is joint with Chinmaya Gupta,
Mark Holland and Andrew Torok.
15.30-16.30 Neil Dobbs, KTH, Stockholm
Phase transitions in unimodal dynamics
Abstract:
We shall give an overview of how phase transtitions can occur in
unimodal dynamics. In particular we shall discuss the influence of
renormalisations on the pressure function.
Tuesday, February 2, 2010:
14.00-15.00 Masato Tsujii, Kyushu University, Fukuoka
Functional analytic methods in smooth ergodic theory (second lecture)
Abstract:
Dynamical zeta functions for Anosov diffeomorphisms
Using the structure of the transfer operator that we show in the first
lecture, we can show some analytic properties of the dynamical zeta
function. The point in the argument is that even though the transfer
operator is not in the trace class, we can define the trace of it and
relate it to the spectral property of the transfer operator.
15.30-16.30 Masato Tsujii. Kyushu University, Fukuoka
Functional analytic methods in smooth ergodic theory (third lecture)
Abstract:
Transfer operators and dynamical zeta function for contact Anosov
flows.
Thursday, January 28, 2010:
14.00-15.00 Ramona Anton, Mathématiques Jussieu, Paris
Global existence for Gross-Pitaevskii equation on three dimensional exterior domains
Abstract:
We prove global existence in the energy space for the Gross-Pitaevskii equation on exterior domains of dimension three. We use a Strichartz estimate adapted to the domain. This estimate follows from a semi-classical dispersive estimate combined with a smoothing effect.
15.30-16.30 Evelina Shamarova, Universidade do Porto
Solutions of Navier-Stokes and Burgers equations via forward-backward SDEs
(joint work with Ana Bela Cruzeiro)
Abstract:
We establish a connection between the strong solution to the spatially periodic Navier-Stokes equations and a solution to a system of forward-backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We describe a construction of $H^s$-regular solutions to the spatially periodic Burgers equation via FBSDEs.
Tuesday, January 26, 2010:
14.00-15.00 Masato Tsujii, Kyushu University, Fukuoka
Functional analytic methods in smooth ergodic theory(a series of three lectures)
Abstract:
(1) Spectrum of transfer operators for Anosov diffeomorphisms
For an Anosov diffeomorphism, we construct what we call "anisotropic
Sobolev space" on the phase space and show that the transfer operator
have nice spectral properties
15.30-16.30 Niklas Brännström, University of Helsinki
Slow dynamics of a slow-fast Hamiltonian system
Abstract:
Hamiltonian dynamics is often used to model problems in mathematical physics. In this talk we will focus on Hamiltonian systems with two time scales, that is, the system has a slow component and a fast component. Integrating the Hamiltonian equations of motion is a hard problem. A view that is often taken is that it is the slow dynamics that represents the dynamics we are interested in while the fast component is just a perturbation (you may think of it as deterministic noise). Then using averaging techniques one may ( terms and conditions apply!) derive approximate slow dynamics. In this talk I will introduce a mechanism with which (under suitable conditions) we can construct orbits which behaves quite differently from those predicted by the averaging theory. Colloquially, I will show a Hamiltonian "write-your-name theorem".
This work has been done in collaboration with V.Gelfreich (Warwick) and E. De Simone (Helsinki).
Thursday, January 21, 2010:
14.00-15.00 Joerg Schmeling, University of Lund
On the dimension of iterated sumsets
Abstract:
Let $A$ be a subset of the real line. We study the fractal dimensions of
the $k$-fold iterated sumsets $kA$, defined as
\[
k A = \{ a_1+ \ldots + a_k : a_i \in A\}.
\]
We show that for any non-decreasing sequence $\{ \alpha_k
\}_{k=1}^\infty$ taking values in $[0,1]$, there exists a compact set
$A$ such that $k A$ has Hausdorff dimension $\alpha_k$ for all $k\ge 1$.
We also show how to control various kinds of dimension simultaneously
for families of iterated sumsets.
These results are in stark contrast to the Pl\"{u}nnecke-Rusza
inequalities in additive combinatorics. However, for lower box-counting
dimension, the analogue of the Pl\"{u}nnecke-Rusza inequalities does hold.
Tuesday, January 19, 2010:
14.00-15.00 Tomas Persson Warsaw
Dimension of piecewise hyperbolic attractors with overlaps
Abstract: We study a general class of piecewise hyperbolic maps on bounded subsets of the plane. Assuming that the map satisfies a transversality condition we prove that the Hausdorff dimension of the attractor is what one might expect from the Lyapunov exponents.
15.30-16.30 Daniel Schnellmann, KTH, Stockholm
Absolutely continuous limit distributions of sums
of point measures
Abstract:
In this talk I will present three results of my recent PhD thesis. In the
first two results, we consider sequences of real numbers in the unit
interval and study how they are distributed. The sequences in the first
paper are given by the forward iterations of a point $x\in[0,1]$ under a
piecewise expanding map $T_a:[0,1]\to[0,1]$ depending on a parameter $a$
contained in an interval $I$. Under the assumption that each $T_a$ admits
a unique absolutely continuous invariant probability measure $\mu_a$ and
that some technical conditions are satisfied, we show that the
distribution of the forward orbit $T_a^j(x)$, $j\ge1$, is described by the
distribution $\mu_a$ for Lebesgue almost every parameter $a\in I$. In the
second result, we apply the ideas of the first paper to certain sequences
which are equidistributed in the unit interval and give a geometrical
proof of an old result by Koksma.\\
In the last result we consider certain Bernoulli convolutions. By showing
that a specific transversality property is satisfied, we deduce absolute
continuity of the to these Bernoulli convolutions associated.
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