ABSTRACTS
Alexander Blokh.
Necessary conditions for the existence of wandering vertices for cubic
polynomials.
Abstract:
For a cubic polynomial with locally connected
Julia set we prove the following necessary
conditions for the existence of wandering
non-precritical branch points of the Julia set:
the Julia set must be a dendrite, the two critical
points must be distinct, at least one of them must
be recurrent, and they must have the same limit set.
Keith Burns.
Partial hyperbolicity, Lyapunov exponents and stable ergodicity
Abstract: The talk will describe
joint work with Yasha Pesin and Dima Dolgpyat. I
will present some results and open problems about stable ergodicity of
partially hyperbolic diffeomorphisms with non-zero Lyapunov exponents. The
main tool is local ergodicity theory for non-uniformly hyperbolic systems.
Jerome Buzzi.
Measures of maximal entropy for surface diffeomorphisms
Abstract: Invariant probability measures with maximal entropy are natural objects
which can be thought of as capturing the "complexity" of the dynamics.
They often have special dynamical properties, e.g., posess "semi-uniform"
hyperbolicity.
In this talk, answering a question of Sheldon Newhouse, we first consider
piecewise affine homeomorphisms of surface with positive topological
entropy and show that they have finitely many ergodic probability measures
with maximal entropy and that these measures are Bernoulli up to a rotation.
We then turn to C^{1+\epsilon}-diffeomorphisms.
Danijela Damjanovic.
An application of KAM to local rigidity of some partially
hyperbolic higher rank actions
Victor Donnay.
The boundary of metrics with ergodic geodesic flow.
Abstract: For metrics with negative Gaussian curvature and focusing
caps, the geodesic flow is ergodic with positive Lyapunov exponents
almost everywhere. There are finitely many periodic orbits that have zero
Lyapunov exponents. Under small perturbations, one can destroy these zero
exponent orbits but at the same time one creates elliptic (KAM stable)
periodic orbits. Thus the perturbed systems are non-ergodic. This result
is similar in spirit to work by Rom-Kudar and Tureav who show that
arbitrarily close to the ergodic non-smooth Sinai billiard are smooth,
non-ergodic billiard systems.
Alex Eskin.
Unipotent flows on branched covers of Veech surfaces
John Franks.
Actions of lattices on low-dimensional manifolds
Abstract: We consider how algebraic properties of discrete subgroups of
Lie groups restrict the possible actions of those groups on
low-dimensional manifolds. The results show a strong parallel between
the possible actions of such a group on the circle S^1 and the
area preserving actions on surfaces.
Jacek Gracyzk.
On the Schwarzian derivative
Abstract: We discuss variuos analytical and metrical properties
of the Schwarzian derivative. In particular, for C^3 interval maps
we prove the existence of an analytic coordinate change which yields
negative Schwarzian.
Carlos Gutierrez.
On the weak Marcus-Yamabe Conjecture
Abstract:
In this talk, we would consider the plain differentiable versions
(not necessarily C^1) of the results of the articles
1) "On local diffeomorphisms of R^n that are injective"
C. Gutierrez and A. Fernandes
http://www.preprint.impa.br/Shadows/SERIE_A/2002/179.html
2) "A solution to the Bidimensional Global Asymptotic Stability
Conjecture"
C. Gutierrez
Ann. Inst. Henri Poincare 12 (1995), 627-671.
The paper (by C. Gutierrez and R. Rabanal) which contains
the results is being prepared.
Eugene Gutkin.
Blocking of orbits and security for polygons
Abstract: Security (or the lack of it) is a dynamical property of billiard
orbits
and geodesics. We investigate the security of polygons and related surfaces.
We prove that among lattice polygons the only secure ones are the
arithmetic polygons.
Corollary: a regular n-gon is secure iff n=3,4,6.
Sandra Hayes.
The dynamics of a Henon map
Abstract: For the Henon family f(x,y) = (y, y^2+ax), 0 < a < 1,
it will be shown that two fixed points are the only real
periodic points and that the basin of attraction of the
origin has the stable manifold of the saddle fixed
point as its boundary. Consider in R^2, these maps are
Morse-Smale diffeomorphisms and consider in C^2,
the basin is a Fatou-Bieberbach domain.
Yulij Ilyashenko.
Minimal attractors and new robust properties of partially hyperbolic
systems
Abstract:A definition of minimal attractor was suggested by the speaker in 1985.
Since then it was proved that the minimal attractor
is responsible for the time averages of continuous functions (Bachourin,
2000). Moreover, if the minimal attractor is
infinite, then the system exibits a mild chaotic behavior (Triple choice
theorem by Gorodetski and Ilyashenko, 1996).
A chalendging problem of the noncoincidence of minimal and Milnor
attractors was investigated recently by V.Kleptsyn.
It appeared that this noncoincidence has in a sense "codimension one
minus epsilon".
New robust properties of dynamical systems were obtained recently by
Gorodetski and the speaker. There are systems
which have simultaneously dense sets of periodic orbits of different
indexes, and a dense orbit with a zero Lyapunov exponent.
The result is based on a concept that any geometric property
observed in a free group action by diffeomorphisms
of a compact manifold may be observed for classical dynamical systems,
namely, diffeomorphisms of a manifold of higher dimension.
An effect of syncronizaton of orbits for free group actions by
diffeomorphisms of a circle was explained by Kleptsyn.
It may be related with the "Fubini nightmare" found by Ruelle, Shub and
Wilkinson.
Michael Jakobson.
Geometric structure of partitions for Henon-like families.
Abstract: Based on the joint work with S. Newhouse we describe
geometric properties of our inductive construction
in the phase space and in the parameter space.
Vadim Kaloshin.
Application of Mather diffusion theorem to instability of
totally elliptic points of symplectic maps in dimension 4.
Anatole Katok.
New progress in measure rigidity of abelian group actions and Littlewood
conjecture in Diophantine approximation
Abstract: This is joint work with M. Einsiedler and E. Lindenstrauss.
Jonathan King.
Limits of powers of a rank-1 transformation, possibilities and restrictions
Dmitry Kleinbock.
Bounded geodesics, slowly recurrent billiard trajectories,
and badly approximable vectors
Abstract: This is a joint work with Barak Weiss. In the moduli
space of quadratic differentials on a Riemann surface
we find many points with bounded Teichmuller geodesic
trajectories. An application to billiards in rational
polygons provides a construction of rational billiard
analogue of badly approximable numbers. Similar methods
show that there exist many badly approximable vectors on
certain fractal subsets of R^n.
Francois Ledrappier.
Distribution results for eigenvalues of
some lattices
Brian Martensen.
Interval Exchange Transformations and Substitution Tiling Spaces
Abstract: (Joint with C. Holton) We show that a (fairly general) tiling
space can embed in a surface if and only if it is the suspension of an
interval exchange transformation. Further, we give a computable and
complete invariant for substitution tiling spaces to embed in a surface.
This results in an algorithm to determine when a substitution is an IET.
Previously, algorithms were only know for substitutions on 2-symbols.
Michal Misiurewicz.
Area preserving attractors
Abstract: We consider a family of locally area preserving branched
coverings of the plane. In particular, if a branching point is
periodic, the structure of the map in the neighbourhood of its orbit
can be investigated using the theory of complex rational maps.
This is a joint work with A. Blokh.
Roman Muchnik.
Stationary measures on hyperbolic spaces
Abstract: We consider a Q-regular measure m on the boundary of a Gromov
hyperbolic space X. If G is a nonelementary discrete group of isometries
of X, we show that under mild conditions there exists a probability
measure on G such that m is stationary. Certain Gibbs states including
the Paterson-Sullivan measures are examples.
We apply this machinery to the case of CAT(-1) spaces to prove that
the measure on G can be chosen to have finite first moment. This proves
that m is Harmonic for the corresponding random walk. In the course of
the proof of the main result, we also show a version of the Shadow Lemma
for delta hyperbolic spaces which are not necessarily geodesic.
Also, for CAT(-1) spaces, we show that Busemann functions
evaluated at orbit points are Lipschitz in the boundary variable with
respect to the Gromov metric(s) and we estimate the Lipschitz constant
We consider a Q-regular measure m on the boundary of a Gromov
hyperbolic space X. If G is a nonelementary discrete group of isometries
of X, we show that under mild conditions there exists a probability
measure on G such that m is stationary. Certain Gibbs states including
the Paterson-Sullivan measures are examples.
We apply this machinery to the case of CAT(-1) spaces to prove that
the measure on G can be chosen to have finite first moment. This proves
that m is Harmonic for the corresponding random walk. In the course of
the proof of the main result, we also show a version of the Shadow Lemma
for delta hyperbolic spaces which are not necessarily geodesic.
Also, for CAT(-1) spaces, we show that Busemann functions
evaluated at orbit points are Lipschitz in the boundary variable with
respect to the Gromov metric(s) and we estimate the Lipschitz constant
Will Ott.
The Distortion of the Dimension Spectra of Fractal Measures Under
Projection
Abstract:
We study the extent to which the Hausdorff dimension and the dimension
spectrum of a fractal measure supported on a compact subset of a Banach
space are affected by a typical mapping into a finite-dimensional
Euclidean space. Previous work establishes that Hausdorff dimension and
an important part of the dimension spectrum are preserved by a typical
mapping when the ambient space is finite-dimensional. A preservation
result for general Banach spaces is too much to hope for. We prove that
for a general Banach space, the Hausdorff dimension and the aforementioned
part of the dimension spectrum are preserved by a typical projection up to
a factor involving the "thickness" of the support of the measure. One
recovers a dimension preservation result if the thickness of the support
of the measure is zero. We conjecture that many of the attractors of the
equations of mathematical physics have thickness zero. This is known to
be true for the two-dimensional incompressible Navier-Stokes system.
This work is
joint with Brian Hunt and Vadim Kaloshin.
Martin Schmoll.
Growth rate functions for degree 2 torus coverings and generalizations
Abstract: We discuss growth rates of geodesic segments for degree two torus
coverings. The parameter space of degree 2 torus coverings contains
surfaces obtained from billiards with a wall.
This simple case has generalizations and applications, moreover the
space of degree 2 torus coverings already contains many arithmetic
surfaces which do not have a "single closed cylinder" direction.
Ilie Ugarcovici.
Chaotic Dynamics in a Discrete Nonlinear Population Model
Abstract: We study the dynamics of an overcompensatory Leslie population model where
the fertility rates decay exponentially with population size. In the two and
three generation versions of this model, through careful numerical studies,
we find a large variety of complicated dynamical behaviors: several types of
codimension-one local bifurcations, two different bifurcation cascades
transforming an attracting invariant closed curve into a strange attractor,
multiple co-existing strange attractors with large basins, crises (interior,
boundary, merging). We show that some of the more exotic phenomena arise
from homoclinic tangencies. Many of these chaotic behaviors have not been
previously observed in population models. This is joint work with Howie
Weiss.
Viorel Nitica.
Topological transitivity of non-compact extensions
Abstract:(Joint with I.Melbourne and A. Torok)
Let K be a compact connected Lie group, R^n the
n-dimensional Euclidean group, and T:M\to M a transitive
Anosov diffeomorphism.
In the set of Holder KxR^n-extensions of T we find a
Holder-open subset of stably transitive transformations, which
is dense in the set of Holder extensions that do not have the
R^n-projection of the periodic weighting separated by
any hyperplane.
We show that in the set of Holder SE(2 m)-extensions of T
there is a Holder-open and dense subset of stably transitive
extensions, which is a slight improvement of a result of Melbourne
and Nicol.
If the base is a shift of finite type, we show that the set of
locally constant SE(2m+1)-extensions contain an open dense
subset of stably transitive extensions.
We show that if the group spanned by a subset S of SE(2m+1) is dense,
then the
semigroup spanned by S is dense.
If the base is a hyperbolic set, we find several criteria for the
transitivity
of the extensions with non-compact fiber. Consequently we find
transitive extensions for any connected Lie group fiber, as well
as open sets of transitive extensions if the fiber has an open set
of elliptic elements close to identity. In particular there are
open sets of stably transitive extensions with fiber
Sp(n,R).
Anna Talitskaya.
On existence of hyperbolic flows on any manifold
Abstract: We show that on any manifold of dimension n>=3
there exists a flow F^t such that for any t\ne 0, F^t has
all but one nonzero Lyapunov exponents and is ergodic.
Christian Wolf.
Measures of maximal dimension for hyperbolic diffeomorphisms
Abstract:We establish the existence of ergodic measures of maximal
Hausdorff dimension for hyperbolic sets of surface
diffeomorphisms. This is a dimension-theoretical version of the
existence of ergodic measures of maximal entropy. The crucial
difference is that while the entropy map is upper-semicontinuous,
the map $\nu\mapsto\dim_H\nu$ is neither upper-semicontinuous nor
lower-semi\-continuous. This forces us to develop a new approach,
which is based on the thermodynamic formalism. Remarkably, for a
generic diffeomorphism with a hyperbolic set, there exists an
ergodic measure of maximal Hausdorff dimension in a particular
two-parameter family of equilibrium measures. This is a joint work
with L. Barreira.
James Yorke.
Pure chaos, A new class of maps in the plane: the movie
Abstract: Examples have played a key role in understanding dynamical systems. Plykin
for example found a superb well-known example of a map in the plane that has
a chaotic attractor on which the dynamics is uniformly hyperbolic, and in
particular is uniformly stretching. The Plykin example has been difficult to
understand, and we show how it can easily be understood through video
animations. The animations lead us to a new class of maps that are uniformly
hyperbolic, smooth, and area preserving (on the invariant set). They have
the nice properties of the piecewise linear "Baker map" but our maps have no
discontinuities. We use computer animations throughout as an aid to
visualization. This is joint work with graduate student J.T. Halbert.
Anton Zorich.
Counting saddle connections on flat surfaces and description
of the "cusps" of the moduli spaces of Abelian differentials.
Abstract: This is joint work with A.Eskin and H.Masur.
A holomorphic 1-form on a compact Riemann surface S naturally
defines a flat metric on S with cone-type singularities. We
present the following surprising phenomenon: having found a
geodesic segment (saddle connection) joining a pair of conical
points one can find with a nonzero probability another saddle
connection on S having the same direction and the same length as
the initial one. A similar phenomenon is valid for the families
of parallel closed geodesics.
We give a description of all possible configurations of parallel
saddle connections (and of families of parallel closed geodesics)
which might be found on a generic flat surface S and we count the
number of corresponding saddle connections of length less than L.
To perform this computation we elaborate the detailed description
of the "cusps" (i.e., of the principal part of the boundary) of
the moduli space of holomorphic 1-forms and we find the volumes
of these "cusps".