Algebraic property of dilatation constants of piecewise linear structures of Anosov foliations
Vol. 11 (2004), No. 1, Page 65--74.
Minakawa, Hiroyuki
Algebraic property of dilatation constants of piecewise linear structures of Anosov foliations
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Abstract:
Let $\phi_t$ be an orientable Anosov flow of a closed 3-manifold and ${\cal F}^s$, the stable foliation of the flow. If ${\cal F}^s$ has a $\lambda$-piecewise linear structure, then we show that it is equivalent to the one obtained by using a surface of section $S$ of the flow. Then we prove, for a positive integer $p$, $\lambda^p$ is equal to the dilatation of the first return mapping of $S$ which is a pseudo-Anosov diffeomorphism of a compact surface with boundary. Therefore, $\lambda$ is a zero of a monic reciprocal polynomial with integral coefficients . In particular, $\lambda$ is not transcendental and this gives a negative answer to the question raised in [10]. We also comment on the Ghys inequality for the group $\text{PL}_{\lambda}(S^1)$ of all orientation preserving piecewise linear homeomorphisms whose derivatives are integral powers of $\lambda$ at each differentiable point.
Mathematics Subject Classification (1991): 58F18, 58F25.
Mathematical Reviews Number: MR2044912
Received: 2003-11-25
