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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.
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Contributor : Jérôme Fehrenbach <>
Submitted on : Monday, January 15, 2007 - 6:28:56 PM
Last modification on : Wednesday, October 14, 2020 - 4:24:21 AM
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  • HAL Id : tel-00124712, version 1



Jérôme Fehrenbach. Quelques apects géométriques et dynamiques du mapping class group. Mathématiques [math]. Université Nice Sophia Antipolis, 1998. Français. ⟨tel-00124712⟩



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