Everything is illuminated (Q309042)

This paper deals with geometrical properties of translation surfaces, namely, the finite blocking and the bounded blocking property and the illumination property. Assuming that \(M\) is a translation surface, a pair of points \((x,y)\in M\times M\) is finitely blocked if there exists a finite set \(B\subset M\) which does not contain \(x\) or \(y\) and intersects every straight-line trajectory connecting \(x\) to \(y.\) The set \(B\) is called a blocking set for \((x,y)\) and the minimal cardinality among all the blocking sets for \((x,y)\) is called the blocking cardinality of \((x,y).\) A translation surface \(M\) has the finite blocking property if any pair \((x,y)\in M\times M\) is finitely blocked, and the bounded blocking property if there is a number \(n\) such that any pair \((x,y)\in M\times M\) is finitely blocked with blocking cardinality at most \(n.\) If the blocking cardinality of a pair \((x,y)\) is zero\ then it is said that \(x\)\ and \(y\)\ do not illuminate each other. In the first theorem of the paper equivalent characterizations of the finite blocking property are given. More precisely, it is proven that the following statements are equivalent: NEWLINENEWLINENEWLINE (1) \(M\) has the finite blocking property; NEWLINENEWLINENEWLINE (2) \(M\) is a torus cover, i.e., there is a translation map from \(M\) to a torus; NEWLINENEWLINENEWLINE (3) there is an open set \(U\subset M\times M\) such that any pair of points in \(U\) is finitely blocked; NEWLINENEWLINENEWLINE (4) \(M\) has the bounded blocking property. NEWLINENEWLINENEWLINE The second result concerns the illumination problem. Results of \textit{P. Hubert} et al. [Int. Math. Res. Not. 2008, Article ID rnn011, 42 p. (2008; Zbl 1156.32006)] are extended by removing the hypothesis that \(M\) is a lattice surface and settling a conjecture of them. In this context, the following theorem is proven: for any translation surface \(M,\) and any point \(x\in M,\) the set of points \(y\) which are not illuminated by \(x\) is finite. Moreover, the set \(\{(x,y):\) \(x\) and \(y\) do not illuminate each other\(\}\) is the union of a finite set with finitely many translation surfaces \(S\) embedded in \(M\times M,\) such that the projections \(p_{i}|_{S}:S\rightarrow M\) are both finite-degree covers of the complement of a finite set in \(M;\) here \(p_{i}:M\times M\rightarrow M,\) \(i=1,2\), are the natural projections onto the first and second factors.NEWLINENEWLINEFinally, the following interesting result is obtained as a corollary of the previous theorem: let \(P\) be a rational polygon. Then for any \(x\in P\) there are at most finitely many points \(y\) for which there is no geodesic trajectory between \(x\) and \(y.\)