


Differentiable Dynamics and Smooth Ergodic Theory
Org: Giovanni Forni and Konstantin Khanin (Toronto) [PDF]
 JAYADEV ATHREYA, Yale University, Dept. of Mathematics, 10 Hillhouse Ave,
New Haven, CT 06511
A lattice point problem in Teichmuller space
[PDF] 
Let T_{g} denote the Teichmuller space of marked compact Riemann
surfaces of genus g. Let G_{g} be the associated mapping class
group. Let Q_{g} be the unit cotangent bundle to T_{g}, that is, the
space of (marked) unit area holomorphic quadratic differentials. Let
p: Q_{g} ® T_{g} be the natural projection, and let
m denote the natural G_{g}invariant measure on Q_{g}. Then
m = p_{*} m is a measure on Teichmuller space. In joint work with
Bufetov, Eskin, and Mirzakhani, we calculate the asymptotics of
 G_{g} x ÈB(y,R), for x,y Î T_{g}, where B(y,R)
denotes the ball of radius R in the Teichmuller metric;
 m ( B(x,R) ).
 PAVEL BATCHOURINE, University of Toronto
On ergodicity of multidimensional dispersing billiards
[PDF] 
Ergodic theory of dispersing billiards was developed in 1970s1980s.
An important part of the theory is the analysis of the structure of
the sets where the billiard map is discontinuous. They were assumed
to be smooth manifolds till recently, when a new pathological type of
behaviour of these sets was found. Thus a reconsideration of earlier
arguments was needed. I'll show that at least in a generic situation
the earlier proofs of ergodicity of dispersing billiards can be
recovered.
 BASSAM FAYAD, Paris 13, av. JB Clément, Villetaneuse, France
Smooth linearization of commuting circle diffeomorphisms
[PDF] 
We show that smooth commuting circle diffeomorphisms with
simultaneously diophantine rotation numbers are smoothly conjugated to
rotations.
 VADIM KALOSHIN, Department of Mathematics, The Pennsylvania State
University, University Park, PA 16802, USA
Hausdorff dimension of oscillatory motions for the 3body
problem
[PDF] 
Consider the classical 3body problem mutually attracted by Newton
gravitation. Call motions oscillatory if at time tends to infinity
limsup of maximal distance among the bodies is infinite, while liminf
is finite. In the '50s Sitnikov presented the first rigorous example
of oscillatory motions for the socalled restricted 3body problem.
Later in the '60s Alexeev extended this example to the 3body problem.
A longstanding conjecture, probably going back to Kolmogorov, is that
oscillatory motions have measure zero. We show that for the Sitnikov
example and for the socalled restricted planar circular 3body
problem these motions often have full Hausdorff dimension.
This is a joint work with Anton Gorodetski.
 FEDERICO RODRIGUEZHERTZ, IMERL, Uruguay
Partial hyperbolicity and ergodicity in dimension three
[PDF] 
In [HHU] we proved the PughShub conjecture for conservative
partially hyperbolic diffeomorphisms with onedimensional center.
That is, stably ergodic diffeomorphisms are dense among the
conservative partially hyperbolic ones. Can we describe this
abundance of ergodicity more accurately?
More precisely:
Problem
Which 3dimensional manifolds support a nonergodic partially
hyperbolic diffeomorphism?
We conjecture that the answer to this question is that the only such
manifolds are the mapping tori of diffeomorphisms commuting with an
Anosov one. In the other cases, being partially hyperbolic would
automatically imply ergodicity. We prove this for a family of
manifolds:
Theorem
Let f : N® N be a conservative partially hyperbolic
C^{2} diffeomorphism where N ¹ T^{3} is a compact
3dimensional nilmanifold. Then, f is ergodic.
Sacksteder [Sa] proved that certain affine diffeomorphisms of
nilmanifolds are ergodic. These examples are partially hyperbolic.
Some of our results apply to other manifolds and we obtain, for
instance, that every conservative partially hyperbolic diffeomorphism
of S^{3} is ergodic but this is probably a theorem about the
empty set.
This is a joint work with María Alejandra Rodriguez Hertz and
Raúl Ures.
References
 [BW]

K. Burnsa and A. Wilkinson,
On the ergodicity of partially hyperbolic systems.
Ann. Math., to appear.
 [PS]

C. Pugh and M. Shub,
Stable ergodicity and julienne quasiconformality.
J. Eur. Math. Soc. 2(2000), 152.
 [HHU]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic
diffeomorphisms with 1dcenter bundle.
Preprint.
 [Sa]

R. Sacksteder,
Strongly mixing transformations.
In: 1970 Global Analysis, Proc. Sympos. Pure Math. XIV,
Berkeley, Calif., 1968, 245252.
 MIKE SHUB, University of Toronto
NonZero Lyapunov Exponents in Families of Dynamical
Systems
[PDF] 
We compare the average Lyapunov exponent in families of twist maps and
one dimensional expanding maps to the Lyapunov exponents of random
products chosen i.i.d. from these families. For some families we
prove that the average exponent is bigger than the random exponent.
In other cases the provable and experimental results are very
suggestive.
Different portions of the work surveyed are joint work with Francois
Ledrappier, Rafael de la LLave, Enrique Pujals, Leonel Robert, Carles
Simo and Amie Wilkinson.
 CORINNA ULCIGRAI, Princeton University, Mathematics Department, Fine Hall,
Washington Rd, Princeton, NJ 08544
Mixing of flows over interval exchange maps
[PDF] 
We consider suspension flows over interval exchange transformations,
under a roof function with logarithmic singularities. As a
motivation, such flows arise as minimal components of flows on
surfaces given by multivalued Hamiltonians. We prove that if the
roof function has an asymmetric logarithmic singularity, the
suspension flow is strongly mixing for a full measure set of interval
exchanges. This generalizes a result by Khanin and Sinai for flows
over rotations of the circle. In the proof we use a recent result by
AvilaGouzelYoccoz.
 AMIE WILKINSON, Northwestern University, Evanston, IL, USA
Asymmetrical diffeomorphisms
[PDF] 
Which diffeomorphisms of a compact manifold M commute with no other
diffeomorphisms (except their own powers)? Smale asked if such highly
asymmetrical diffeomorphisms are typical, in that they are dense in
the C^{r} topology on the space of C^{r} diffeomorphisms Diff^{r}(M).
In this talk I will explain the recent (positive) solution to Smale's
question for C^{1} symplectomorphisms and volumepreserving
diffeomorphisms. I will also discuss progress on the general
case.
This is joint work with Christian Bonatti and Sylvain Crovisier.

