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

Lanneau, Erwan; Thiffeault, Jean-Luc

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 $2g$.

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