41 research outputs found
Periodic points of Hamiltonian surface diffeomorphisms
The main result of this paper is that every non-trivial Hamiltonian
diffeomorphism of a closed oriented surface of genus at least one has periodic
points of arbitrarily high period. The same result is true for S^2 provided the
diffeomorphism has at least three fixed points. In addition we show that up to
isotopy relative to its fixed point set, every orientation preserving
diffeomorphism F: S --> S of a closed orientable surface has a normal form. If
the fixed point set is finite this is just the Thurston normal form.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper20.abs.htm
On train track splitting sequences
We show that the subsurface projection of a train track splitting sequence is
an unparameterized quasi-geodesic in the curve complex of the subsurface. For
the proof we introduce induced tracks, efficient position, and wide curves.
This result is an important step in the proof that the disk complex is Gromov
hyperbolic. As another application we show that train track sliding and
splitting sequences give quasi-geodesics in the train track graph, generalizing
a result of Hamenstaedt [Invent. Math.].Comment: 40 pages, 12 figure
Entropy zero area preserving diffeomorphisms of
In this paper we formulate and prove a structure theorem for area preserving
diffeomorphisms of genus zero surfaces with zero entropy. As an application we
relate the existence of faithful actions of a finite index subgroup of the
mapping class group of a closed surface on by area preserving
diffeomorphisms to the existence of finite index subgroups of bounded mapping
class groups with non-trivial first cohomology.Comment: 88 pages, 4 figure
Uniform Hyperbolicity of the Graphs of Curves
Let denote the curve complex of the closed orientable
surface of genus with punctures. Masur-Minksy and subsequently Bowditch
showed that is -hyperbolic for some
. In this paper, we show that there exists some
independent of such that the curve graph is
-hyperbolic. Furthermore, we use the main tool in the proof of this
theorem to show uniform boundedness of two other quantities which a priori grow
with and : the curve complex distance between two vertex cycles of the
same train track, and the Lipschitz constants of the map from Teichm\"{u}ller
space to sending a Riemann surface to the curve(s) of shortest
extremal length.Comment: 19 pages, 2 figures. This is a second version, revised to fix minor
typos and to make the end of the main proof more understandabl
Completely reducible sets
International audienceWe study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets