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Theses
Quelques apects géométriques et dynamiques du mapping class group
Quelques apects géométriques et dynamiques du mapping class group
Abstract : Let S be a compact oriented surface equipped with n+1 distinguished points. In the first chapter of this thesis, we recall the theory of efficient representatives for a pseudo-Anosov element of the mapping class group of S. This objects have been introduced by Bestvina-Handel and Los.
In the second chapter, we expose the theory of good representatives and super efficient representatives of a pseudo-Anosov map fixing the puncture x_0. We show that the set of super efficient representatives has a special structure : it is a union of a finite number of cycles, the cycles are looped around when applying some elementary combinatorial operations. We derive from this result algorithms to decide if a given homeomorphism f - or its isotopy class - admits a root fixing x_0, or commutes with a finite order homeomorphism fixing x_0. We also give a new solution to the conjugacy problem among pseudo-Anosov elements of the mapping class group fixing x_0.
In the third chapter, we consider a homeomorphism f of the disk and O a periodic orbit for f of period n>=3. We give a lower bound for the topological entropy of the homeomorphisms isotopic to f relatively to O. This estimation is obtained using the theory of efficient representatives.
In the fourth chapter, we give necessary and sufficient conditions for a given n-string braid beta to admit a destabilization or an exchange move. These conditions are expressed in terms of the element of the mapping class group induced by the braid beta.
In the second chapter, we expose the theory of good representatives and super efficient representatives of a pseudo-Anosov map fixing the puncture x_0. We show that the set of super efficient representatives has a special structure : it is a union of a finite number of cycles, the cycles are looped around when applying some elementary combinatorial operations. We derive from this result algorithms to decide if a given homeomorphism f - or its isotopy class - admits a root fixing x_0, or commutes with a finite order homeomorphism fixing x_0. We also give a new solution to the conjugacy problem among pseudo-Anosov elements of the mapping class group fixing x_0.
In the third chapter, we consider a homeomorphism f of the disk and O a periodic orbit for f of period n>=3. We give a lower bound for the topological entropy of the homeomorphisms isotopic to f relatively to O. This estimation is obtained using the theory of efficient representatives.
In the fourth chapter, we give necessary and sufficient conditions for a given n-string braid beta to admit a destabilization or an exchange move. These conditions are expressed in terms of the element of the mapping class group induced by the braid beta.
https://tel.archives-ouvertes.fr/tel-00124712
Contributor : Jérôme Fehrenbach <>
Submitted on : Monday, January 15, 2007 - 6:28:56 PM
Last modification on : Wednesday, October 14, 2020 - 4:24:21 AM
Long-term archiving on: : Wednesday, April 7, 2010 - 2:12:30 AM
Contributor : Jérôme Fehrenbach <>
Submitted on : Monday, January 15, 2007 - 6:28:56 PM
Last modification on : Wednesday, October 14, 2020 - 4:24:21 AM
Long-term archiving on: : Wednesday, April 7, 2010 - 2:12:30 AM