362 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
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,w) where C is a smooth compact complex
curve of a given genus and w is a holomorphic 1-form on C with a given list of
multiplicities of zeroes. We describe connected components of this space.
This classification is important in the study of dynamics of interval
exchange transformations and billiards in rational polygons, and in the study
of geometry of translation surfaces.Comment: 42 pages, 12 figures, LaTe
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
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
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