(Organizers: Joe Auslander,
Mike Boyle, Giovanni
Forni)
Date
Speaker (Affiliation)
Title/Abstract
January 26
NO SEMINAR
---
January 31 NOTE: The seminar is on Tuesday this week.
Sam Senti (Institute of Mathematics of the Federal
University of Rio de Janeiro)
Title:
Thermodynamic formalism for the Henon map at the first bifurcation
Abstract:
We study the dynamics of strongly dissipative Henon maps, at the first
bifurcation parameter where the uniform hyperbolicity is destroyed by
the formation of tangencies inside the limit set. We prove the existence
of an equilibrium measure which minimizes the free energy associated
with the non continuous potential -t\log J^u, where t is in a certain
interval of the form (-\infty, t_0), t_0>0 and J^u denotes the Jacobian
in the unstable direction. We also prove the occurrence of a phase
transition at which multiple equilibrium measures coexist and the
pressure function is not differentiable. This is a joint work with
Hiroki Takahasi from the University of Tokyo
Title:
The invariant measures of some infinite interval exchange
maps
Abstract: A construction of Thurston produces produces
a translation surface from a graph and a positive eigenfunction
for the adjacency operator. The construction produces surfaces with
interesting symmetry groups.
I will consider surfaces built in this way from infinite graphs.
Fixing such a surface and a direction in [0,2*pi), we can study
the unit speed flow in this direction. For most such surfaces and
some directions, I will explain a characterization
of the locally finite ergodic invariant measures for this flow. These
ergodic measures are in bijective correspondence with the extremal
positive eigenfunctions of the graph. In many cases, this characterization
can be promoted to a classification of ergodic invariant measures
utilizing known Martin boundary theory for the graph.
Title:
Part I). Interval exchange transformations. Part II).
Quantitative shrinking targets for IETs and rotations.
(For a pdf file compiling the tex abstract below, click
here.)
Abstract:
Part I). Interval exchange transformations are invertible, piecewise
order preserving isometries of the unit interval with finitely many
discontinuities. Starting from rotations of the circle, which they
generalize, this talk will present their connections to flows on flat
surfaces, rational billiards and symbolic coding. Recent results on
diophantine approximation for interval exchange transformations will be
presented.
Abstract:
Part II). In this talk we present some quantitative shrinking target
results. Consider $T:[0,1] \to [0,1]$. One can ask how quickly under T a
typical point $x$ approaches a typical point $y$. In particular given
$\{a_i\}_{i=1}^{\infty}$ is $T^ix \in B(y,a_i)$ infinitely often? A finer
question of whether $T^ix \in B(y,a_i)$ as often as one would expect will
be discussed. That is, does
$$\underset{N \to \infty}{\lim}\frac{\underset{n=1}{\overset{N}{\sum}}
\chi_{B(y,a_n)(T^nx)}}{\underset{n=1}{\overset{N}{\sum}} 2a_n}=1$$
for almost every $x$.
We will present applications to billiards in rational polygons and a
related result for Sturmian sequences. This is joint work with David
Constantine.
Title:
Stationary Markov random fields and Gibbs measures
Abstract:
Markov random fields are higher-dimensional analogs of Markov-chains,
expressing a conditional independence property. The
Hammersley-Clifford theorem states that Markov Random fields are Gibbs
measures with a nearest neighbor interaction, under an assumption on
the support: the existence of a ``safe symbol''. In this talk I will
present joint work (in progress) with Nishant Chandgotia, Guangyue
Han, Brian Marcus and Ronnie Pavlov, investigating Markov random fields.
March 1
Vadim Kaloshin (Univ. of Maryland)
Title:
On conjugacy of convex billiards
(joint with A.
Sorrentino)
Abstract:
We show that if two billiard maps of convex domains are
C^2-conjugate near
the boundary, then the corresponding domains are similar, i.e. they
can be
obtained one from the other by a rescaling and an isometry. As an
application,
we prove a conditional version of the Birkhoff conjecture on the
integrability of planar
billiards and show that the original conjecture is equivalent to
what we call
an Extension problem. Quite interestingly, our result and a positive
solution to
this extension problem would provide an answer to a closely related
question
in spectral theory: if the marked length spectra of two domains are
the same,
is it true that they are isometric?
March 8
Room 3206
Marcel Guardia
(U. Politecnica de Catalunya)
This is a joint meeting with the PDE seminar.
The talk will be in the Colloquium Room, Room 3206.
Title:
Growth of Sobolev norms for the cubic defocusing nonlinear Schroedinger equation in polynomial time
Abstract:
For the abstract on the PDE seminar page, click
here.
March 15
Kelly Funk (University of Illinois at Urbana-Champaign)
Title: On Rigidity Sequences
Abstract:
In this talk we will discuss rigidity sequences and uniform rigidity
sequences. I will give examples of each for weakly mixing transformations and
inform you of what has been done thus far to characterize the structure of
these sequences. Then I will prove a generic result regarding weakly mixing
homeomorphisms of the two torus that are uniformly rigid and show that if a
sequence satisfies a certain growth rate then there is a weakly mixing
homeomorphism of the two torus that is uniformly rigid with respect to the
given sequence.
Title: A criterion for the simplicity of the Lyapunov spectrum of
square-tiled surfaces (joint with M. Moeller and J.-C. Yoccoz)
Abstract:
The Lyapunov exponents of the so-called Kontsevich-Zorich (KZ) cocycle
over the Teichmuller geodesic flow are important quantities associated to the
study of deviation of ergodic averages of interval exchange transformations,
translation flows and billiards.
After the seminal works of G. Forni, and A. Avila and M Viana on the so-called
Kontsevich-Zorich conjecture, we know that the Lyapunov exponents of KZ cocycle
with respect to the natural absolutely continuous (Masur-Veech) probability are
simple (i.e., they are non-zero and their multiplicity is 1).
On the other hand, G. Forni (and his collaborators) constructed particular
examples of probabilities associated to square-tiled surfaces whose Lyapunov
spectrum for the KZ cocycle have zero and/or non-simple exponents.
In particular, one can ask whether there is a (say, sufficient) criterion to
decide the simplicity of the Lyapunov spectrum associated to square-tiled
surfaces. In this talk we will discuss a joint work with Martin Moller and
Jean-Christophe Yoccoz showing such a simplicity criterion (in the spirit of the
work of A. Avila and M. Viana). As a consequence of this simplicity criterion,
we will exhibit an infinity family of square-tiled surfaces (of genus 3) of the
minimal stratum H(4) meeting the conditions of our simplicity criterion.
Title:
Partially hyperbolic systems close to trivial extensions (with
C. Liverani)
Abstract:
We consider a class of partially hyperbolic systems on the two-dimensional
torus given by smooth $\varepsilon$-perturbations of maps
$F(x,\theta)=(f(x,\theta),\theta)$ where $f(\cdot,\theta)$ are smooth
expanding maps of the circle. For sufficiently small $\varepsilon$ we prove
existence and uniqueness of a SRB measure and exponential decay of correlation
for H\"older observables with exponentially small rate
$\exp(-c/\varepsilon)$; if additionally $f(x,\theta)=f(x)$ does not depend on
$\theta$, the system presents exponential decay of correlation with rate
$-\varepsilon/\log\varepsilon$.
Title:
Limit theorems for toral translations
Abstract:
We study the discrepancy of the number of visits of a Kronecker sequence on T^d
to nice sets. We are interested in particular in the question how the
answer depends on the geometry of the set. It is a joint work with Bassam Fayad.
Title:
Smoothing Lyapunov functions
Abstract:
This is a joint work Pierre Pageault.
A Lyapunov function is a a function which is non-increasing along orbits of a
dynamical systems. Continuous Lyapunov functions for flows have been
constructed by Conley.
Smooth Lyapunov functions can be constructed by Conley's method for
homeomorphisms. However there are examples of continuous flows which do not
admit a smooth non-constant Lyapunov function. We will explain this phenomenon.
We will also give results on the possibility of approximating a
Lyapunov function by a smooth Lyapunov function
April 26
Livio Flaminio (Lille)
Title:
Flows Cohomologies and equi-distribution
Abstract:
For nilflows on Heisenberg 3-maniflods, in a joint work with Forni, we gave
accurate estimates of Birkhoff averages by studying the dynamics
of "renormalization" on the bundle of degree 1 cohomology of the moduli
space of nilflow.
These methods generalize to other situations.
On the one hand, by considering the cohomology in higher degree one
can estimate Birkhoff averages for action of R^n on (2n +1)-Heisenberg manifolds.
(Joint work with S. Cosentino)
On the other hand, even in the absence of a true dynamic
renormalization, it is possible to obtain
Quantitative estimates of Birkhoff averages
for almost any initial point for the flows on
nilmanifolds of particular groups of greater degree
of nilpotency. (Joint work with G. Forni)
The talk will focus on the second topic above.
Title:
Unique equilibrium states for certain robustly transitive
diffeomorphisms
Abstract:
(Joint work with Vaughn Climenhaga and Dan Thompson)
During the first hour of the seminar we will discuss the concepts of
topological pressure and equilibrium states. We will also outline a
classical argument due to Bowen that expansive systems with
specification have unique equilibrium states. During the second hour
we will explain recent generalizations to Bowen's argument that show
the existence of equilibrium states for a larger class of systems. We
then apply these results to certain robustly transitive
diffeomorphisms.
Title:
Uncovering the Bifurcation Structure of the Diblock Copolymer Model
Abstract:
The diblock copolymer equation is a fundamental model for phase separation
processes which involve long-range interactions, and therefore promote the
formation of fine structure. From a mathematical point of view, the
evolution
equation arises from the classical Cahn-Hilliard model by the addition of
a linear term, which is due to the addition of a nonlocal term to the
associated
energy. While the equilibrium structure of the Cahn-Hilliard model on
one-dimensional
domains is fully understood, a complete description of the diblock copolymer
equilibrium structure is still unknown. In this talk we describe some
preliminary
results which describe the formation of energy minimizers with fine
structure
through a homotopy from the classical Cahn-Hilliard bifurcation diagram, as
well
as related multistability issues.
This talk is based on joint work with Ian Johnson and Thomas Wanner.
Geodesics on Margulis Spacetimes Abstract:
A Margulis spacetime is a 3-manifold which is a quotient of 3-space by
a free group of affine transformations.
Associated to every such 3-manifold M is a hyperbolic surface S.
Generalizing the correspondence between closed geodesics on M
and closed geodesics on S, we establish an orbit equivalence
between recurrent spacelike geodesics on $M^3$ and
recurrent geodesics on S. In contrast, no timelike
geodesic recurs in either forward or backwards time.
This is joint work with Francois Labourie.
September 22
Joseph Auslander (UMCP)
Regional Proximality, McMahon's Theorem, and the Veech Relation
September 29
Abed Bounemoura (IAS)
Persistence of invariant submanifolds
Abstract: In this talk, we will give a simple and geometrical proof of
the classical result on persistence and uniqueness for normally
hyperbolic submanifolds. Moreover, the proof gives a new result on
persistence for a wider class of submanifolds, which we call
topologically normally hyperbolic. This is a joint work with Pierre
Berger.
October 6
Pierre Pageault (ENS Lyon)
Functions whose set of critical points is an arc
Abstract: We prove that on a compact connected manifold M with
dim(M)>1, the set of C^1 functions whose set of critical points is an
arc is dense for the C^0 topology. We then present applications in
dynamic, and link them with uniqueness problems of Weak KAM solution
associated to Mañé Lagrangians.
October 13
Mark Demers (Fairfield)
A spectral gap for the transfer operator of the Lorentz gas
Abstract: Much attention has been given in recent years to developing a
framework to study directly the transfer operator associated with
hyperbolic maps on an appropriate Banach space. For the billiard map
associated with a Lorentz gas of both finite and infinite horizon, we
construct generalized function spaces on which the transfer operator is
quasi-compact and has a spectral gap. This framework gives a unified
approach to proving the statistical properties and various limit laws
associated with billiards, such as exponential decay of correlations,
central limit theorem and large deviation estimates. It also has
potential applications to many classes of perturbations. This is joint
work Hong-Kun Zhang.
October 20
NO SEMINAR
Semi-annual Workshop in Dynamical Systems and Related Topics, Penn St.
October 27
Nikita Selinger (SUNY Stony Brook)
Title: The proof of Pilgrim's conjecture.
Abstract: Let $f$ be a postcritically finite branched self-cover of a
2-dimensional topological sphere. Such a map induces an analytic self-map
$\sigma_f$ of a finite-dimensional Teichm\"uller space. We
prove that this map extends continuously to the
augmented Teichm\"uller space and give an explicit construction for
this extension. This allows us to characterize the dynamics of
Thurston's pullback map near invariant strata of the boundary of the
augmented Teichm\"uller space. The resulting classification of
invariant boundary strata is used to prove a conjecture by Pilgrim.
November 3
Alex Eskin (Chicago) Note: This speaker will also give the department colloquium on November 2.
Sums of Lyapunov exponents of the Teichmueller geodesic flow, and the
Siegel-Veech constants This is joint work with Maxim Kontsevich and Anton Zorich
November 10
Francesco Cellarosi (IAS)
Title: Random square-free numbers.
Abstract: A square-free number is an integer that is not divisible by p^2 for any prime p. I shall discuss two ways of generating 'random' square-free numbers. One construction is inspired by Statistical Mechanics and enjoys some unexpected properties, such as a non-standard limit theorem. The second construction is more classical and can be understood using a 'natural' dynamical system, whose ergodic properties have been recently examined by P. Sarnak in connection with his conjecture on the randomness for the Möbius function. Joint work with Ya.G. Sinai.
November 17
David Aulicino (UMCP)
Title: "Classifying Teichm\”uller Disks with Completely Degenerate Kontsevich-Zorich Spectrum"
Abstract:
The moduli space of genus g Riemann surfaces is the space of all complex structures on a closed orientable surface of genus g up to orientation preserving diffeomorphisms. The Teichm\"uller geodesic flow is the flow on the cotangent bundle of the Teichm\"uller space of surfaces defined by the direction of minimal dilatation and it descends to the cotangent bundle of
the moduli space under the action of the mapping class group. It is well-known that the Lyapunov spectrum of this flow is determined by g numbers 1=\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_g \geq 0. The Kontsevich-Zorich conjecture, proven by Forni and Avila-Viana, showed
that generically all the inequalities are strict with respect to the canonical absolutely continuous measures. However, Forni found an example of a measure on the genus three moduli space, and Forni-Matheus found a measure in genus four, with completely degenerate spectrum, i.e. 1=\lambda_1 > \lambda_2 = \cdots = \lambda_g=0. We prove that these are the only such measures in genus three and four. Furthermore, there are no such measures for g=2 and g \geq 13. Finally, if there are no square-tiled surfaces in genus five that determine a measure with completely degenerate spectrum, then there
are no examples for g \geq 5.
November 24
NO SEMINAR
THANKSGIVING BREAK
December 1
Barney Bramham (IAS) Please note that this seminar is joint with the Geometry/Topology seminar.
Title: "Approximating Hamiltonian systems by integrable systems using pseudo-holomorphic curves".
Abstract: I will talk about an approach, using pseudo-holomorphic curve
techniques from symplectic geometry, to the following question in dynamical systems of Anatole Katok: "In low dimensions is every conservative dynamical system with zero topological entropy a limit of integrable systems?"
December 8
Robbie Robinson (George Washington)
Title: Kakaya's theorem and maps of the interval
Abstract: In 1924 Soichi Kakeya described a common generalization of decimal and continued fraction expansions. The idea was reinvented by Bissinger and Everett in the 1940s,
and in 1957 Renyi recast the idea in terms of maps of the interval, where it is now called f-expansions. In this talk we will discuss Kakeya's original theorem on the subject in the context of interval maps, as well discussing some contributions of Renyi, Parry and others.
