The Geometric Approach for Computing the Joint Spectral Radius

This research was supported by the grant RFBR 05-01-00066 and by the grant 304.2003.1 supporting the leading scientific schools. In this paper we describe the geometric approach for computing the joint spectral radius of a finite family of linear operators acting in finite-dimensional Eucledian space. The main idea is to use the invariant sets of of these operators. It is shown that any irreducible family of operators possesses a centrally-symmetric invariant compact set, not necessarily unique. The Minkowski norm generated by the convex hull of an invariant set (invariant body) possesses special extremal properties that can be put to good use in exploring the joint spectral radius. In particular, approximation of the invariant bodies by polytopes gives an algorithm for computing the joint spectral radius with a prescribed relative deviation ε. This algorithm is polynomial with respect to 1/ε if the dimension is fixed. Another direction of our research is the asymptotic behavior of the orbit of an arbitrary point under the action of all products of given operators. We observe some relations between the constants of the asymptotic estimations and the sizes of the invariant bodies. In the last section we give a short overview on the extension of geometric approach to the Lp-spectral radius.