On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus

Lanneau, Erwan; Thiffeault, Jean-Luc

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, Erwan ; Thiffeault, Jean-Luc

Annales de l'Institut Fourier, Tome 61 (2011), p. 105-144 / Harvested from Numdam

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Résumé
Abstract

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 $2 g$ .

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 $2 g$ .

Publié le : 2011-01-01

MR 2828128 | Zbl 1237.37027 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2599
Classification: 37D40, 37E30
Mots clés: homéomorphisme de type pseudo-Asanov, petite dilatation, surface

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     author = {Lanneau, Erwan and Thiffeault, Jean-Luc},
     title = {On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus},
     journal = {Annales de l'Institut Fourier},
     volume = {61},
     year = {2011},
     pages = {105-144},
     doi = {10.5802/aif.2599},
     zbl = {1237.37027},
     mrnumber = {2828128},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2011__61_1_105_0}
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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) pp. 105-144. doi : 10.5802/aif.2599. http://gdmltest.u-ga.fr/item/AIF_2011__61_1_105_0/

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