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Continuous dependence of geodesic frames in the Hausdorff metric. (English) Zbl 1087.37039
Math. Notes 77, No. 6, 862-864 (2005); translation from Mat. Zametki 77, No. 6, 935-937 (2005).
Let us consider the closed orientable surface \(M^2\) of genus \(\leq 2\) and let \(L(M^2)\) denote the set of orientable geodesic laminations on \(M^2\).
In this note, the author defines a metric on \(L(M^2)\) using the Hausdorff metric on the space of closed subsets of the tangent bundle \(T(M^2)\).
For a geodesic lamination \(\lambda\subset L(M^2)\), the \(\widehat\lambda\) denotes its lifting to \(T(M^2)\). The Hausdorff distance between geodesics laminations \(\lambda,\lambda'\subset L(M^2)\) is defined by \(d_H(\lambda,\lambda')= \widehat d_H(\widehat\lambda,\widehat\lambda')\) where \(\widehat d_H(A,B)\) is Hausdorff distance between closed sets \(A,B\subset T(M^2)\). Then, the author shows, that the geodesic frames of irrational flows with structurally stable rest points have the same properties as the irrational Poincaré rotation numbers of minimal flows on the torus, namely: 1. continuity (in the Hausdorff metric) with respect to the perturbation of the flow; 2. “instability” – the initial irrational frame can be made rational by an arbitrarily small perturbation.
MSC:
37E35 Flows on surfaces
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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