153 research outputs found
Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory
The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory
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
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