# American Institute of Mathematical Sciences

• Previous Article
Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces
• DCDS Home
• This Issue
• Next Article
Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate
September  2016, 36(9): 4871-4893. doi: 10.3934/dcds.2016010

## Generic absence of finite blocking for interior points of Birkhoff billiards

 1 Department of Mathematics, Indiana University, Bloomington, IN 47405, United States, United States

Received  August 2015 Revised  March 2016 Published  May 2016

Let $x$ and $y$ be points in a billiard table $M=M(\sigma)$ in $\mathbb{\mathbb{R}}^{2}$ that is bounded by a curve $\sigma$. We assume $\sigma\in\Sigma_{r}$ with $r\geq2$, where $\Sigma_{r}$ is the set of simple closed $C^{r}$ curves in $\mathbb{R}^{2}$ with positive curvature. A subset $B$ of $M\setminus\{x,y\}$ is called a blocking set for the pair $(x,y)$ if every billiard path in $M$ from $x$ to $y$ passes through a point in $B$. If a finite blocking set exists, the pair $(x,y)$ is called secure in $M;$ if not, it is called insecure. We show that for $\sigma$ in a dense $G_{\delta}$ subset of $\Sigma_{r}$ with the $C^{r}$ topology, there exists a dense $G_{\delta}$ subset $\mathcal{\mathcal{R}=R}(\sigma)$ of $M(\sigma)\times M(\sigma)$ such that $(x,y)$ is insecure in $M(\sigma)$ for each $(x,y)\in\mathcal{R}$. In this sense, for the generic Birkhoff billiard, the generic pair of interior points is insecure. This is related to a theorem of S. Tabachnikov [24] that $(x,y)$ is insecure for all sufficiently close points $x$ and $y$ on a strictly convex arc on the boundary of a smooth table.
Citation: Thomas Dauer, Marlies Gerber. Generic absence of finite blocking for interior points of Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4871-4893. doi: 10.3934/dcds.2016010
##### References:
 [1] V. Bangert and E. Gutkin, Insecurity for compact surfaces of positive genus,, Geom. Dedicata, 146 (2010), 165.  doi: 10.1007/s10711-009-9432-8.  Google Scholar [2] R. Bishop, Circular billiard tables, conjugate loci, and a cardiod,, Regul. Chaotic Dyn., 8 (2003), 83.  doi: 10.1070/RD2003v008n01ABEH000227.  Google Scholar [3] J. Bruce and P. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory,, Cambridge University Press, (1984).   Google Scholar [4] K. Burns and M. Gidea, Differential Geometry and Topology: With a View to Dynamical Systems,, Chapman & Hall/CRC, (2005).   Google Scholar [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.  doi: 10.3934/dcds.2008.21.403.  Google Scholar [6] M. Farber, Topology of Billiard Problems, I,, Duke Math J., 115 (2002), 559.  doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar [7] M. Farber, Topology of Billiard Problems, II,, Duke Math J., 115 (2002), 587.  doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar [8] M. Gerber and L. Liu, Real analytic metrics on $S^{2}$ with total absence of finite blocking,, Geom. Dedicata, 166 (2013), 99.  doi: 10.1007/s10711-012-9787-0.  Google Scholar [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.  doi: 10.4310/MRL.2011.v18.n3.a1.  Google Scholar [10] E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Ergodic Theory Dynam. Sys., 4 (1984), 569.  doi: 10.1017/S0143385700002650.  Google Scholar [11] E. Gutkin, Blocking of billiard orbits and security for polygons and flat surfaces,, Geom. Funct. Anal., 15 (2005), 83.  doi: 10.1007/s00039-005-0502-2.  Google Scholar [12] E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems,, Chaos, 22 (2012).  doi: 10.1063/1.4729307.  Google Scholar [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.  doi: 10.1016/j.ansens.2003.05.001.  Google Scholar [14] E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards,, Math. Res. Lett., 3 (1996), 391.  doi: 10.4310/MRL.1996.v3.n3.a8.  Google Scholar [15] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [16] E. Gutkin and V. Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces,, Geom. Dedicata, 118 (2006), 185.  doi: 10.1007/s10711-005-9036-x.  Google Scholar [17] W. Ho, On blocking numbers of surfaces,, preprint, (2008).   Google Scholar [18] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar [19] J.-F. Lafont and B. Schmidt, Blocking light in compact Riemannian manifolds,, Geom. Topol., 11 (2007), 867.  doi: 10.2140/gt.2007.11.867.  Google Scholar [20] T. Monteil, A counter-example to the theorem of Hiemer and Snurnikov,, J. Statist. Phys., 114 (2004), 1619.  doi: 10.1023/B:JOSS.0000013974.81162.20.  Google Scholar [21] J. Oxtoby, Measure and Category, Second Edition,, Springer-Verlag, (1980).   Google Scholar [22] W. Rudin, Principles of Mathematical Analysis, Third Edition,, McGraw Hill, (1976).   Google Scholar [23] S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005).  doi: 10.1090/stml/030.  Google Scholar [24] S. Tabachnikov, Birkhoff billiards are insecure,, Discrete Contin. Dyn. Syst., 23 (2009), 1035.  doi: 10.3934/dcds.2009.23.1035.  Google Scholar [25] W. Veech, Teichmüller curves in moduli space, Eisenstein series, and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar [26] W. Veech, The billiard in a regular polygon,, Geom. Funct. Anal., 2 (1992), 341.  doi: 10.1007/BF01896876.  Google Scholar [27] Ya. Vorobets, On the measure of the set of periodic points of a billiard,, Math. Notes, 55 (1994), 455.  doi: 10.1007/BF02110371.  Google Scholar [28] M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents,, Comm. Math. Phys., 105 (1986), 391.  doi: 10.1007/BF01205934.  Google Scholar

show all references

##### References:
 [1] V. Bangert and E. Gutkin, Insecurity for compact surfaces of positive genus,, Geom. Dedicata, 146 (2010), 165.  doi: 10.1007/s10711-009-9432-8.  Google Scholar [2] R. Bishop, Circular billiard tables, conjugate loci, and a cardiod,, Regul. Chaotic Dyn., 8 (2003), 83.  doi: 10.1070/RD2003v008n01ABEH000227.  Google Scholar [3] J. Bruce and P. Giblin, Curves and Singularities: A Geometrical Introduction to Singularity Theory,, Cambridge University Press, (1984).   Google Scholar [4] K. Burns and M. Gidea, Differential Geometry and Topology: With a View to Dynamical Systems,, Chapman & Hall/CRC, (2005).   Google Scholar [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.  doi: 10.3934/dcds.2008.21.403.  Google Scholar [6] M. Farber, Topology of Billiard Problems, I,, Duke Math J., 115 (2002), 559.  doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar [7] M. Farber, Topology of Billiard Problems, II,, Duke Math J., 115 (2002), 587.  doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar [8] M. Gerber and L. Liu, Real analytic metrics on $S^{2}$ with total absence of finite blocking,, Geom. Dedicata, 166 (2013), 99.  doi: 10.1007/s10711-012-9787-0.  Google Scholar [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.  doi: 10.4310/MRL.2011.v18.n3.a1.  Google Scholar [10] E. Gutkin, Billiards on almost integrable polyhedral surfaces,, Ergodic Theory Dynam. Sys., 4 (1984), 569.  doi: 10.1017/S0143385700002650.  Google Scholar [11] E. Gutkin, Blocking of billiard orbits and security for polygons and flat surfaces,, Geom. Funct. Anal., 15 (2005), 83.  doi: 10.1007/s00039-005-0502-2.  Google Scholar [12] E. Gutkin, Billiard dynamics: An updated survey with the emphasis on open problems,, Chaos, 22 (2012).  doi: 10.1063/1.4729307.  Google Scholar [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.  doi: 10.1016/j.ansens.2003.05.001.  Google Scholar [14] E. Gutkin and C. Judge, The geometry and arithmetic of translation surfaces with applications to polygonal billiards,, Math. Res. Lett., 3 (1996), 391.  doi: 10.4310/MRL.1996.v3.n3.a8.  Google Scholar [15] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191.  doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar [16] E. Gutkin and V. Schroeder, Connecting geodesics and security of configurations in compact locally symmetric spaces,, Geom. Dedicata, 118 (2006), 185.  doi: 10.1007/s10711-005-9036-x.  Google Scholar [17] W. Ho, On blocking numbers of surfaces,, preprint, (2008).   Google Scholar [18] A. Katok and B. Hasselblatt, Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar [19] J.-F. Lafont and B. Schmidt, Blocking light in compact Riemannian manifolds,, Geom. Topol., 11 (2007), 867.  doi: 10.2140/gt.2007.11.867.  Google Scholar [20] T. Monteil, A counter-example to the theorem of Hiemer and Snurnikov,, J. Statist. Phys., 114 (2004), 1619.  doi: 10.1023/B:JOSS.0000013974.81162.20.  Google Scholar [21] J. Oxtoby, Measure and Category, Second Edition,, Springer-Verlag, (1980).   Google Scholar [22] W. Rudin, Principles of Mathematical Analysis, Third Edition,, McGraw Hill, (1976).   Google Scholar [23] S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005).  doi: 10.1090/stml/030.  Google Scholar [24] S. Tabachnikov, Birkhoff billiards are insecure,, Discrete Contin. Dyn. Syst., 23 (2009), 1035.  doi: 10.3934/dcds.2009.23.1035.  Google Scholar [25] W. Veech, Teichmüller curves in moduli space, Eisenstein series, and an application to triangular billiards,, Invent. Math., 97 (1989), 553.  doi: 10.1007/BF01388890.  Google Scholar [26] W. Veech, The billiard in a regular polygon,, Geom. Funct. Anal., 2 (1992), 341.  doi: 10.1007/BF01896876.  Google Scholar [27] Ya. Vorobets, On the measure of the set of periodic points of a billiard,, Math. Notes, 55 (1994), 455.  doi: 10.1007/BF02110371.  Google Scholar [28] M. Wojtkowski, Principles for the design of billiards with nonvanishing Lyapunov exponents,, Comm. Math. Phys., 105 (1986), 391.  doi: 10.1007/BF01205934.  Google Scholar
 [1] Serge Tabachnikov. Birkhoff billiards are insecure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035 [2] Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055 [3] Krzysztof Frączek, Ronggang Shi, Corinna Ulcigrai. Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions. Journal of Modern Dynamics, 2018, 12: 55-122. doi: 10.3934/jmd.2018004 [4] Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121 [5] Piotr Kościelniak, Marcin Mazur. On $C^0$ genericity of various shadowing properties. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523 [6] Neal Koblitz, Alfred Menezes. Another look at security definitions. Advances in Mathematics of Communications, 2013, 7 (1) : 1-38. doi: 10.3934/amc.2013.7.1 [7] Isabelle Déchène. On the security of generalized Jacobian cryptosystems. Advances in Mathematics of Communications, 2007, 1 (4) : 413-426. doi: 10.3934/amc.2007.1.413 [8] Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623 [9] Zoltán Buczolich, Balázs Maga, Ryo Moore. Generic Birkhoff spectra. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6649-6679. doi: 10.3934/dcds.2020131 [10] D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 843-876. doi: 10.3934/dcds.2005.13.843 [11] Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65 [12] W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159 [13] Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893 [14] Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255 [15] Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048 [16] Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020319 [17] Alex Eskin, Maryam Mirzakhani. Counting closed geodesics in moduli space. Journal of Modern Dynamics, 2011, 5 (1) : 71-105. doi: 10.3934/jmd.2011.5.71 [18] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051 [19] Palash Sarkar, Subhadip Singha. Verifying solutions to LWE with implications for concrete security. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020057 [20] Riccardo Aragona, Alessio Meneghetti. Type-preserving matrices and security of block ciphers. Advances in Mathematics of Communications, 2019, 13 (2) : 235-251. doi: 10.3934/amc.2019016

2019 Impact Factor: 1.338