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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, 2016, 36 (9) : 4871-4893. doi: 10.3934/dcds.2016010
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show all references

References:
[1]

Geom. Dedicata, 146 (2010), 165-191. doi: 10.1007/s10711-009-9432-8.  Google Scholar

[2]

Regul. Chaotic Dyn., 8 (2003), 83-95. doi: 10.1070/RD2003v008n01ABEH000227.  Google Scholar

[3]

Cambridge University Press, Cambridge, 1984.  Google Scholar

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Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 21 (2008), 403-413. doi: 10.3934/dcds.2008.21.403.  Google Scholar

[6]

Duke Math J., 115 (2002), 559-585. doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar

[7]

Duke Math J., 115 (2002), 587-621. doi: 10.1215/S0012-7094-02-11535-X.  Google Scholar

[8]

Geom. Dedicata, 166 (2013), 99-128. doi: 10.1007/s10711-012-9787-0.  Google Scholar

[9]

Math. Res. Lett., 18 (2011), 389-404. doi: 10.4310/MRL.2011.v18.n3.a1.  Google Scholar

[10]

Ergodic Theory Dynam. Sys., 4 (1984), 569-584. doi: 10.1017/S0143385700002650.  Google Scholar

[11]

Geom. Funct. Anal., 15 (2005), 83-105. doi: 10.1007/s00039-005-0502-2.  Google Scholar

[12]

Chaos, 22 (2012), 026116, 13pp. doi: 10.1063/1.4729307.  Google Scholar

[13]

Ann. Sci. École Norm. Sup. (4), 36 (2003), 847-866. doi: 10.1016/j.ansens.2003.05.001.  Google Scholar

[14]

Math. Res. Lett., 3 (1996), 391-403. doi: 10.4310/MRL.1996.v3.n3.a8.  Google Scholar

[15]

Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3.  Google Scholar

[16]

Geom. Dedicata, 118 (2006), 185-208. doi: 10.1007/s10711-005-9036-x.  Google Scholar

[17]

preprint, arXiv:0807.2934v3 (2008). Google Scholar

[18]

Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[19]

Geom. Topol., 11 (2007), 867-887. doi: 10.2140/gt.2007.11.867.  Google Scholar

[20]

J. Statist. Phys., 114 (2004), 1619-1623. doi: 10.1023/B:JOSS.0000013974.81162.20.  Google Scholar

[21]

Springer-Verlag, New York-Berlin, 1980.  Google Scholar

[22]

McGraw Hill, New York-Auckland-D\"usseldorf, 1976.  Google Scholar

[23]

American Mathematical Society, Providence, RI, 2005. doi: 10.1090/stml/030.  Google Scholar

[24]

Discrete Contin. Dyn. Syst., 23 (2009), 1035-1040. doi: 10.3934/dcds.2009.23.1035.  Google Scholar

[25]

Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar

[26]

Geom. Funct. Anal., 2 (1992), 341-379. doi: 10.1007/BF01896876.  Google Scholar

[27]

Math. Notes, 55 (1994), 455-460. doi: 10.1007/BF02110371.  Google Scholar

[28]

Comm. Math. Phys., 105 (1986), 391-414. doi: 10.1007/BF01205934.  Google Scholar

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