no. 1
On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus
[Dilatation minimales des homéomorphismes de type pseudo-Anosov sur des surfaces de petit genres]
Lanneau, Erwan1 ; Thiffeault, Jean-Luc2
1 Université du Sud Toulon-Var and Fédération de Recherches des Unités de Mathématiques de Marseille Centre de Physique Théorique (CPT) UMR CNRS 6207,Luminy, Case 907 13288 Marseille Cedex 9 (France)
2 University of Wisconsin Department of Mathematics Van Vleck Hall, 480 Lincoln Drive Madison, WI 53706 (USA)
Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 105-144.

Nous calculons la plus petite dilatation d’un homéomorphisme de type pseudo-Anosov laissant invariant un feuilletage mesuré orientable sur une surface de genre g pour g=3,4,5. Nous donnons aussi une borne inférieure pour les genres 6,7 et 8. Nos techniques simplifient la preuve de Cho et Ham sur le calcul de la plus petite dilatation d’un homéomorphisme de type pseudo-Anosov sur une surface de genre 2. Pour g=2 à 5, la plus petite dilatation est le plus petit nombre de Salem pour les polynomes à degré fixé 2g.

We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham’s proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surface. For genus g=2 to 5, the minimum dilatation is the smallest Salem number for polynomials of degree 2g.

DOI : 10.5802/aif.2599
Classification : 37D40, 37E30
Keywords: Pseudo-Anosov homeomorphism, small dilatation, flat surface
Mot clés : homéomorphisme de type pseudo-Asanov, petite dilatation, surface

Lanneau, Erwan 1 ; Thiffeault, Jean-Luc 2

1 Université du Sud Toulon-Var and Fédération de Recherches des Unités de Mathématiques de Marseille Centre de Physique Théorique (CPT) UMR CNRS 6207,Luminy, Case 907 13288 Marseille Cedex 9 (France)
2 University of Wisconsin Department of Mathematics Van Vleck Hall, 480 Lincoln Drive Madison, WI 53706 (USA) 
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     title = {On the minimum dilatation of {pseudo-Anosov} homeromorphisms on surfaces of small genus},
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Lanneau, Erwan; Thiffeault, Jean-Luc. On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus. Annales de l'Institut Fourier, Tome 61 (2011) no. 1, pp. 105-144. doi : 10.5802/aif.2599. https://aif.centre-mersenne.org/articles/10.5802/aif.2599/

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