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Transverse intersections of foliations in three-manifolds. (English) Zbl 1026.57023

Ghys, Étienne (ed.) et al., Essays on geometry and related topics. Mémoires dédiés à André Haefliger. Vol. 2. Genève: L’Enseignement Mathématique. Monogr. Enseign. Math. 38, 503-525 (2001).
The main theorems of the paper are the following:
1. Let \(A\) be an element of \(\lim \operatorname {SL}( 2,\mathbb Z)\) such that \(|\lim\operatorname {Tr} A|>2\). Let \(M\) be the torus bundle over the circle with the monodromy matrix \(A\). On \(M\), there is the natural Anosov flow which is the suspension of the toral automorphism induced by \(A\), and this flow is accompanied by the unstable foliation \(\mathcal F^u\) and the stable foliation \(\mathcal F^s\). Let \(h\) be a diffeomorphism of \(M\) isotopic to the identity and let \(\mathcal F^u\) and \(h^{\ast}\mathcal F^s\) intersect transversely. Then there is a diffeomorphism \(\widehat h\) of \(M\) isotopic to the identity such that \(\widehat h^{\ast}\mathcal F^u=\mathcal F^u\) and \(\widehat h^{\ast}\mathcal F^s=h^{\ast}\mathcal F^s\). Thus, the intersection \(\mathcal F^u\cap h^{\ast}\mathcal F^s\) is isotopic to the suspension Anosov flow \(\mathcal F^u\cap\mathcal F^s\).
2. Let \(\mathcal F^u\) and \(\mathcal F^s\) be the unstable foliation and the stable foliation of the geodesic flow on the unit tangent bundle \(M\) of a closed surface \(\Sigma\) of genus greater than 1. There exists a diffeomorphism \(h\) isotopic to the identity such that the foliations \(\mathcal F^u\) and \(h^{\ast}\mathcal F^s\) are transverse and the multifoliation \((\mathcal F^u,h^{\ast}\mathcal F^s)\) is not isomorphic to \((\mathcal F^u,\mathcal F^s)\).
For the entire collection see [Zbl 0988.00115].

MSC:

57R30 Foliations in differential topology; geometric theory
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
57N10 Topology of general \(3\)-manifolds (MSC2010)
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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