Generic absence of finite blocking for interior points of Birkhoff billiards

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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

Thomas Dauer ^1, and Marlies Gerber ^1,

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

Received August 2015 Revised March 2016 Published May 2016

Abstract
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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.

Keywords: geodesics, genericity., security, Birkhoff billiards, finite blocking.

Mathematics Subject Classification: Primary: 37J99, 37E99, 78A05; Secondary: 53C2.

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.

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