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Generic absence of finite blocking for interior points of Birkhoff billiards

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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.
    Mathematics Subject Classification: Primary: 37J99, 37E99, 78A05; Secondary: 53C22.


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