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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

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