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Spiraling spectra of geodesic lines in negatively curved manifolds. (English) Zbl 1228.53055

Math. Z. 268, No. 1-2, 101-142 (2011); erratum ibid. 276, 3-4, 1215-1216 (2014).
The authors define a “spiraling spectrum” for each closed geodesic in a negatively curved geodesic metric space. This describes the tendency of geodesics to stay away from a closed geodesic for a long time before being drawn in rapidly toward the closed geodesic. They prove, among other things, that the spiraling spectrum is bounded and that if the metric space is a constant curvature manifold, then the spiraling spectrum is closed. Further, if the constant curvature manifold is of dimension at least 3, then the spiraling spectrum contains a closed interval \([0,c]\) for some \(c>0\). They use these results to obtain approximation theorems for real or complex numbers by quadratic irrationals.

MSC:

53C22 Geodesics in global differential geometry
11J06 Markov and Lagrange spectra and generalizations
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
11J83 Metric theory
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