Abstract
Using a well-known example of Livshits, for every \({n > 2}\) we construct obstacles K in \({\mathbb{R}^n}\) such that the set of trapped points for the billiard flow in the exterior of K has a non-empty interior, and therefore a positive Lebesgue measure.
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References
Hirsch M.: Differential Topology. Springer, Berlin (1976)
L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. III, Springer, Berlin, 1985.
Lax P., Phillips R.: Scattering Theory. Academic Press, Amsterdam (1967)
Melrose R.: Geometric Scattering Theory. Cambridge University Press, Cambridge (1995)
Melrose R., Sjöstrand J.: Singularities in boundary value problems I. Comm. Pure Appl. Math. 31, 593–617 (1978)
Melrose R., Sjöstrand J.: Singularities in boundary value problems II. Comm. Pure Appl. Math. 35, 129–168 (1982)
Noakes L., Stoyanov L.: Rigidity of scattering lengths and traveling times for disjoint unions of convex bodies. Proc. Amer. Math. Soc. 143, 3879–3893 (2015)
Noakes L., Stoyanov L.: Traveling times in scattering by obstacles. J. Math. Anal. Appl. 430, 703–717 (2015)
V. Petkov and L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, John Wiley & Sons, Chichester, 1992.
Stoyanov L.: Generalized Hamiltonian flow and Poisson relation for the scattering kernel. Ann. Sci. École. Norm. Sup. 33, 361–382 (2000)
Stoyanov L.: Rigidity of the scattering length spectrum. Math. Ann. 324, 743–771 (2002)
L. Stoyanov, Stability of trapping sets of positive measure, Preprint, 2016.
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Noakes, L., Stoyanov, L. Obstacles with non-trivial trapping sets in higher dimensions. Arch. Math. 107, 73–80 (2016). https://doi.org/10.1007/s00013-016-0907-1
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DOI: https://doi.org/10.1007/s00013-016-0907-1