50 research outputs found
Geodesics on Flat Surfaces
This short survey illustrates the ideas of Teichmuller dynamics. As a model
application we consider the asymptotic topology of generic geodesics on a
"flat" surface and count closed geodesics and saddle connections. This survey
is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.Comment: (25 pages, 5 figures) Based on the talk at ICM 2006 at Madrid; see
Proceedings of the ICM, Madrid, Spain, 2006, EMS, 121-146 for the final
version. For a more detailed survey see the paper "Flat Surfaces",
arXiv.math.DS/060939
Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera
We state conjectures on the asymptotic behavior of the volumes of moduli
spaces of Abelian differentials and their Siegel-Veech constants as genus tends
to infinity. We provide certain numerical evidence, describe recent advances
and the state of the art towards proving these conjectures.Comment: Some background material is added on request of the referee. To
appear in Arnold Math. Journa
Zero Lyapunov exponents of the Hodge bundle
By the results of G. Forni and of R. Trevi\~no, the Lyapunov spectrum of the
Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and
of quadratic differentials does not contain zeroes even though for certain
invariant submanifolds zero exponents are present in the Lyapunov spectrum. In
all previously known examples, the zero exponents correspond to those
PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy
of the Gauss-Manin connection acts by isometries of the Hodge metric. We
present an example of an arithmetic Teichm\"uller curve, for which the real
Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles,
and nevertheless its spectrum of Lyapunov exponents contains zeroes. We
describe the mechanism of this phenomenon; it covers the previously known
situation as a particular case. Conjecturally, this is the only way zero
exponents can appear in the Lyapunov spectrum of the Hodge bundle for any
PSL(2,R)-invariant probability measure.Comment: 47 pages, 10 figures. Final version (based on the referee's report).
A slightly shorter version of this article will appear in Commentarii
Mathematici Helvetici. A pdf file containing a copy of the Mathematica
routine "FMZ3-Zariski-numerics_det1.nb" is available at this link here:
http://w3.impa.br/~cmateus/files/FMZ3-Zariski-numerics_det1.pd
Flat Surfaces
Various problems of geometry, topology and dynamical systems on surfaces as
well as some questions concerning one-dimensional dynamical systems lead to the
study of closed surfaces endowed with a flat metric with several cone-type
singularities. Such flat surfaces are naturally organized into families which
appear to be isomorphic to the moduli spaces of holomorphic one-forms.
One can obtain much information about the geometry and dynamics of an
individual flat surface by studying both its orbit under the Teichmuller
geodesic flow and under the linear group action. In particular, the Teichmuller
geodesic flow plays the role of a time acceleration machine (renormalization
procedure) which allows to study the asymptotic behavior of interval exchange
transformations and of surface foliations.
This long survey is an attempt to present some selected ideas, concepts and
facts in Teichmuller dynamics in a playful way.Comment: (152 pages; 51 figures) Based on the lectures given by the author at
the Les Houches School "Number Theory and Physics" in March of 2003 and at
the workshop on dynamical systems in ICTP, Trieste, in July 2004. See
"Frontiers in Number Theory, Physics and Geometry. Volume 1: On random
matrices, zeta functions and dynamical systems'', P.Cartier; B.Julia;
P.Moussa; P.Vanhove (Editors), Springer-Verlag (2006) for the entire
collection (including, in particular, the complementary lectures of J.-C.
Yoccoz). For a short version see the paper "Geodesics on Flat Surfaces",
arXiv.math.GT/060939
Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants
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. The similar phenomenon is valid for the families of parallel closed
geodesics.
We give a complete 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. We count the number of saddle connections
of length less than L on a generic flat surface S; we also count the number of
admissible configurations of pairs (triples,...) of saddle connections; we
count the analogous numbers of configurations of families of closed geodesics.
By the previous result of A.Eskin and H.Masur these numbers have quadratic
asymptotics with respect to L. Here we explicitly compute the constant in this
quqadratic asymptotics for a configuration of every type. The constant is found
from a Siegel--Veech formula.
To perform this computation we elaborate the detailed description of the
principal part of the boundary of the moduli space of holomorphic 1-forms and
we find the numerical value of the normalized volume of the tubular
neighborhood of the boundary. We use this for evaluation of integrals over the
moduli space.Comment: Corrected typos, modified some proofs and pictures; added a journal
referenc
Lyapunov spectrum of square-tiled cyclic covers
A cyclic cover over the Riemann sphere branched at four points inherits a
natural flat structure from the "pillow" flat structure on the basic sphere. We
give an explicit formula for all individual Lyapunov exponents of the Hodge
bundle over the corresponding arithmetic Teichmuller curve. The key technical
element is evaluation of degrees of line subbundles of the Hodge bundle,
corresponding to eigenspaces of the induced action of deck transformations.Comment: The presentation is simplified. The algebro-geometric background is
described more clearly and in more details. Some typos are correcte