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Generic absence of finite blocking for interior points of Birkhoff billiards
1. | Department of Mathematics, Indiana University, Bloomington, IN 47405, United States, United States |
References:
[1] |
V. Bangert and E. Gutkin, Insecurity for compact surfaces of positive genus, Geom. Dedicata, 146 (2010), 165-191.
doi: 10.1007/s10711-009-9432-8. |
[2] |
R. Bishop, Circular billiard tables, conjugate loci, and a cardiod, Regul. Chaotic Dyn., 8 (2003), 83-95.
doi: 10.1070/RD2003v008n01ABEH000227. |
[3] |
J. Bruce and P. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, Cambridge University Press, Cambridge, 1984. |
[4] |
K. Burns and M. Gidea, Differential Geometry and Topology: With a View to Dynamical Systems, Chapman & Hall/CRC, Boca Raton, FL, 2005. |
[5] |
K. Burns and E. Gutkin, Growth of the number of geodesics between points and insecurity for Riemannian manifolds, Discrete Contin. Dyn. Syst., 21 (2008), 403-413.
doi: 10.3934/dcds.2008.21.403. |
[6] |
M. Farber, Topology of Billiard Problems, I, Duke Math J., 115 (2002), 559-585.
doi: 10.1215/S0012-7094-02-11535-X. |
[7] |
M. Farber, Topology of Billiard Problems, II, Duke Math J., 115 (2002), 587-621.
doi: 10.1215/S0012-7094-02-11535-X. |
[8] |
M. Gerber and L. Liu, Real analytic metrics on $S^{2}$ with total absence of finite blocking, Geom. Dedicata, 166 (2013), 99-128.
doi: 10.1007/s10711-012-9787-0. |
[9] |
M. Gerber and W.-K. Ku, A dense G-delta set of Riemannian metrics without the finite blocking property, Math. Res. Lett., 18 (2011), 389-404.
doi: 10.4310/MRL.2011.v18.n3.a1. |
[10] |
E. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dynam. Sys., 4 (1984), 569-584.
doi: 10.1017/S0143385700002650. |
[11] |
E. Gutkin, Blocking of billiard orbits and security for polygons and flat surfaces, Geom. Funct. Anal., 15 (2005), 83-105.
doi: 10.1007/s00039-005-0502-2. |
[12] |
E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems, Chaos, 22 (2012), 026116, 13pp.
doi: 10.1063/1.4729307. |
[13] |
E. Gutkin, P. Hubert and T. Schmidt, Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity, Ann. Sci. École Norm. Sup. (4), 36 (2003), 847-866.
doi: 10.1016/j.ansens.2003.05.001. |
[14] |
E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403.
doi: 10.4310/MRL.1996.v3.n3.a8. |
[15] |
E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213.
doi: 10.1215/S0012-7094-00-10321-3. |
[16] |
E. Gutkin and V. Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces, Geom. Dedicata, 118 (2006), 185-208.
doi: 10.1007/s10711-005-9036-x. |
[17] |
W. Ho, On blocking numbers of surfaces, preprint, arXiv:0807.2934v3 (2008). |
[18] |
A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[19] |
J.-F. Lafont and B. Schmidt, Blocking light in compact Riemannian manifolds, Geom. Topol., 11 (2007), 867-887.
doi: 10.2140/gt.2007.11.867. |
[20] |
T. Monteil, A counter-example to the theorem of Hiemer and Snurnikov, J. Statist. Phys., 114 (2004), 1619-1623.
doi: 10.1023/B:JOSS.0000013974.81162.20. |
[21] |
J. Oxtoby, Measure and Category, Second Edition, Springer-Verlag, New York-Berlin, 1980. |
[22] |
W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw Hill, New York-Auckland-D\"usseldorf, 1976. |
[23] |
S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/stml/030. |
[24] |
S. Tabachnikov, Birkhoff billiards are insecure, Discrete Contin. Dyn. Syst., 23 (2009), 1035-1040.
doi: 10.3934/dcds.2009.23.1035. |
[25] |
W. Veech, Teichmüller curves in moduli space, Eisenstein series, and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.
doi: 10.1007/BF01388890. |
[26] |
W. Veech, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379.
doi: 10.1007/BF01896876. |
[27] |
Ya. Vorobets, On the measure of the set of periodic points of a billiard, Math. Notes, 55 (1994), 455-460.
doi: 10.1007/BF02110371. |
[28] |
M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys., 105 (1986), 391-414.
doi: 10.1007/BF01205934. |
show all references
References:
[1] |
V. Bangert and E. Gutkin, Insecurity for compact surfaces of positive genus, Geom. Dedicata, 146 (2010), 165-191.
doi: 10.1007/s10711-009-9432-8. |
[2] |
R. Bishop, Circular billiard tables, conjugate loci, and a cardiod, Regul. Chaotic Dyn., 8 (2003), 83-95.
doi: 10.1070/RD2003v008n01ABEH000227. |
[3] |
J. Bruce and P. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory, Cambridge University Press, Cambridge, 1984. |
[4] |
K. Burns and M. Gidea, Differential Geometry and Topology: With a View to Dynamical Systems, Chapman & Hall/CRC, Boca Raton, FL, 2005. |
[5] |
K. Burns and E. Gutkin, Growth of the number of geodesics between points and insecurity for Riemannian manifolds, Discrete Contin. Dyn. Syst., 21 (2008), 403-413.
doi: 10.3934/dcds.2008.21.403. |
[6] |
M. Farber, Topology of Billiard Problems, I, Duke Math J., 115 (2002), 559-585.
doi: 10.1215/S0012-7094-02-11535-X. |
[7] |
M. Farber, Topology of Billiard Problems, II, Duke Math J., 115 (2002), 587-621.
doi: 10.1215/S0012-7094-02-11535-X. |
[8] |
M. Gerber and L. Liu, Real analytic metrics on $S^{2}$ with total absence of finite blocking, Geom. Dedicata, 166 (2013), 99-128.
doi: 10.1007/s10711-012-9787-0. |
[9] |
M. Gerber and W.-K. Ku, A dense G-delta set of Riemannian metrics without the finite blocking property, Math. Res. Lett., 18 (2011), 389-404.
doi: 10.4310/MRL.2011.v18.n3.a1. |
[10] |
E. Gutkin, Billiards on almost integrable polyhedral surfaces, Ergodic Theory Dynam. Sys., 4 (1984), 569-584.
doi: 10.1017/S0143385700002650. |
[11] |
E. Gutkin, Blocking of billiard orbits and security for polygons and flat surfaces, Geom. Funct. Anal., 15 (2005), 83-105.
doi: 10.1007/s00039-005-0502-2. |
[12] |
E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems, Chaos, 22 (2012), 026116, 13pp.
doi: 10.1063/1.4729307. |
[13] |
E. Gutkin, P. Hubert and T. Schmidt, Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity, Ann. Sci. École Norm. Sup. (4), 36 (2003), 847-866.
doi: 10.1016/j.ansens.2003.05.001. |
[14] |
E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards, Math. Res. Lett., 3 (1996), 391-403.
doi: 10.4310/MRL.1996.v3.n3.a8. |
[15] |
E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213.
doi: 10.1215/S0012-7094-00-10321-3. |
[16] |
E. Gutkin and V. Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces, Geom. Dedicata, 118 (2006), 185-208.
doi: 10.1007/s10711-005-9036-x. |
[17] |
W. Ho, On blocking numbers of surfaces, preprint, arXiv:0807.2934v3 (2008). |
[18] |
A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
doi: 10.1017/CBO9780511809187. |
[19] |
J.-F. Lafont and B. Schmidt, Blocking light in compact Riemannian manifolds, Geom. Topol., 11 (2007), 867-887.
doi: 10.2140/gt.2007.11.867. |
[20] |
T. Monteil, A counter-example to the theorem of Hiemer and Snurnikov, J. Statist. Phys., 114 (2004), 1619-1623.
doi: 10.1023/B:JOSS.0000013974.81162.20. |
[21] |
J. Oxtoby, Measure and Category, Second Edition, Springer-Verlag, New York-Berlin, 1980. |
[22] |
W. Rudin, Principles of Mathematical Analysis, Third Edition, McGraw Hill, New York-Auckland-D\"usseldorf, 1976. |
[23] |
S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/stml/030. |
[24] |
S. Tabachnikov, Birkhoff billiards are insecure, Discrete Contin. Dyn. Syst., 23 (2009), 1035-1040.
doi: 10.3934/dcds.2009.23.1035. |
[25] |
W. Veech, Teichmüller curves in moduli space, Eisenstein series, and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.
doi: 10.1007/BF01388890. |
[26] |
W. Veech, The billiard in a regular polygon, Geom. Funct. Anal., 2 (1992), 341-379.
doi: 10.1007/BF01896876. |
[27] |
Ya. Vorobets, On the measure of the set of periodic points of a billiard, Math. Notes, 55 (1994), 455-460.
doi: 10.1007/BF02110371. |
[28] |
M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents, Comm. Math. Phys., 105 (1986), 391-414.
doi: 10.1007/BF01205934. |
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