Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard
The author studies the stability of periodic 2-orbits of the billiard dynamics on a higher dimensional convex table \(\mathcal{D}\) and provides several viewpoints from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics). Note that the caustics of a billiard table \(\mathcal{D}\) consists of the focal points of \(\Gamma=\partial\mathcal{D}\) with respect to points \(z\in\Gamma\). Moreover, the author gives an estimate of the dimension of the unstable subspace at these orbits in terms of the geometry of the caustic and the Morse indices of the length function. More precisely, let \(z\) be a point on the boundary \(\Gamma\) and \(I(z)\) be the segment on \(\mathcal{D}\) intersecting \(\Gamma\) orthogonally at \(z\). Denote by \(L(z)\) the length of the segment \(I(z)\). Then Theorem 1 states that, under some mild conditions, the dimension of the unstable subspace at a periodic 2-orbit equals to the Morse index. Theorem 2 calculates the characteristic polynomial of the Poincaré operator as the determinant of a matrix of only half the dimension. Hill's formula is also generalized to the higher dimensional case. Sufficient conditions are given to ensure that all multipliers lie on the unit circle.
scientific article; zbMATH DE number 5963534