• 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 and Continuous Dynamical Systems, 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), 165-191. 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.

[1]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035

[2]

Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2005, 12 (3) : 523-530. doi: 10.3934/dcds.2005.12.523

[6]

Luigi Chierchia, Gabriella Pinzari. Planetary Birkhoff normal forms. Journal of Modern Dynamics, 2011, 5 (4) : 623-664. doi: 10.3934/jmd.2011.5.623

[7]

Zoltán Buczolich, Balázs Maga, Ryo Moore. Generic Birkhoff spectra. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6649-6679. doi: 10.3934/dcds.2020131

[8]

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

[9]

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

[10]

D. G. Aronson, N. V. Mantzaris, Hans Othmer. Wave propagation and blocking in inhomogeneous media. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893

[14]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[15]

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

[16]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[17]

Balázs Bárány, Michaƚ Rams, Ruxi Shi. On the multifractal spectrum of weighted Birkhoff averages. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2461-2497. doi: 10.3934/dcds.2021199

[18]

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

[19]

Margarida Camarinha, Fátima Silva Leite, Peter Crouch. Riemannian cubics close to geodesics at the boundaries. Journal of Geometric Mechanics, 2022  doi: 10.3934/jgm.2022003

[20]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (148)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]