Differentiable Dynamics and Smooth Ergodic Theory
Org: Giovanni Forni and Konstantin Khanin (Toronto)
- JAYADEV ATHREYA, Yale University, Dept. of Mathematics, 10 Hillhouse Ave,
New Haven, CT 06511
A lattice point problem in Teichmuller space
Let Tg denote the Teichmuller space of marked compact Riemann
surfaces of genus g. Let Gg be the associated mapping class
group. Let Qg be the unit co-tangent bundle to Tg, that is, the
space of (marked) unit area holomorphic quadratic differentials. Let
p: Qg ® Tg be the natural projection, and let
m denote the natural Gg-invariant measure on Qg. Then
m = p* m is a measure on Teichmuller space. In joint work with
Bufetov, Eskin, and Mirzakhani, we calculate the asymptotics of
- |Gg x ÈB(y,R)|, for x,y Î Tg, 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
Ergodic theory of dispersing billiards was developed in 1970s-1980s.
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
- BASSAM FAYAD, Paris 13, av. JB Clément, Villetaneuse, France
Smooth linearization of commuting circle diffeomorphisms
We show that smooth commuting circle diffeomorphisms with
simultaneously diophantine rotation numbers are smoothly conjugated to
- VADIM KALOSHIN, Department of Mathematics, The Pennsylvania State
University, University Park, PA 16802, USA
Hausdorff dimension of oscillatory motions for the 3-body
Consider the classical 3-body 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 so-called restricted 3-body problem.
Later in the '60s Alexeev extended this example to the 3-body problem.
A long-standing conjecture, probably going back to Kolmogorov, is that
oscillatory motions have measure zero. We show that for the Sitnikov
example and for the so-called restricted planar circular 3-body
problem these motions often have full Hausdorff dimension.
This is a joint work with Anton Gorodetski.
- FEDERICO RODRIGUEZ-HERTZ, IMERL, Uruguay
Partial hyperbolicity and ergodicity in dimension three
In [HHU] we proved the Pugh-Shub conjecture for conservative
partially hyperbolic diffeomorphisms with one-dimensional center.
That is, stably ergodic diffeomorphisms are dense among the
conservative partially hyperbolic ones. Can we describe this
abundance of ergodicity more accurately?
Which 3-dimensional manifolds support a non-ergodic partially
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
Let f : N® N be a conservative partially hyperbolic
C2 diffeomorphism where N ¹ T3 is a compact
3-dimensional 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 S3 is ergodic but this is probably a theorem about the
This is a joint work with María Alejandra Rodriguez Hertz and
K. Burnsa and A. Wilkinson,
On the ergodicity of partially hyperbolic systems.
Ann. Math., to appear.
C. Pugh and M. Shub,
Stable ergodicity and julienne quasiconformality.
J. Eur. Math. Soc. 2(2000), 1-52.
F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures,
Accessibility and stable ergodicity for partially hyperbolic
diffeomorphisms with 1d-center bundle.
Strongly mixing transformations.
In: 1970 Global Analysis, Proc. Sympos. Pure Math. XIV,
Berkeley, Calif., 1968, 245-252.
- MIKE SHUB, University of Toronto
Non-Zero Lyapunov Exponents in Families of Dynamical
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
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
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 multi-valued 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
- AMIE WILKINSON, Northwestern University, Evanston, IL, USA
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 Cr topology on the space of Cr diffeomorphisms Diffr(M).
In this talk I will explain the recent (positive) solution to Smale's
question for C1 symplectomorphisms and volume-preserving
diffeomorphisms. I will also discuss progress on the general
This is joint work with Christian Bonatti and Sylvain Crovisier.