Examples of pseudo-Anosov homeomorphisms with small dilatations

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 2, Page 95--111.

Minakawa, Hiroyuki
Examples of pseudo-Anosov homeomorphisms with small dilatations
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Abstract:
For a closed orientable surface $\Sigma_g$ of genus $g\ge 2$, we give an upper bound for the least dilatation of pseudo-Anosov homeomorphisms of $\Sigma_g$. For this purpose, we construct a family of Birkhoff sections for a suspension Anosov flow. Birkhoff sections have been constructed by using cut and paste arguments for surfaces in 3-manifolds. But we construct them directly and then we obtain piecewise linear models of the first return mappings of the Birkhoff sections. These models enable us to investigate their dilatations explicitly, and we obtain necessary estimates.

Mathematics Subject Classification (2000): Primary 37D20; Secondary 37E30, 57N05
Mathematical Reviews Number: MR2277516

Received: 2005-05-12