The homogenization of an ordinary differential equation with a chessboard structure (Q810703)

Let f(x,y) denote the function taking value \(\alpha\) (resp. \(\beta\)) on black (resp. white) squares of the infinite chessboard and let y denote any solution of the differential equation \(y'=f(x,y)\). It is known from a theorem of L. Piccinini that \(\gamma (\alpha,\beta)=\lim_{x\to \infty}y(x)/x\) exists and is independent of y. In this note, the authors show that the function \(\gamma\) is continuous and increasing but very intricated, despite of the linearity of f in \(\alpha\) and \(\beta\). The more remarkable result is the description of the contour line \(\gamma (\alpha,\beta)=c\) that highly depends on arithmetic properties of c. Proofs are not given but ``will appear in a forthcoming paper''.