On the joint spectral radius

G. Panti, Pythagorean triples, billiards, and mean free paths. We show that primitive pythagorean triples can be enumerated by playing billiard on a mgonal billiard table in the Poincaré disk. The resulting first-return billiard map B is a (m− 1)-to-1 orientation-reversing covering map of the circle, a property shared by the group character T (z) = z−(m−1). We prove that there exists a homeomorphism Φ, unique up to postcomposition with elements in a dihedral group, that conjugates B with T . This homeomorphism —whose prototype is the Minkowski question mark function— establishes a bijection between the set of points of degree ≤ 2 over Q(i) and the torsion subgroup of the circle. We prove that Φ is singular and Hölder continuous with exponent log(m− 1) divided by the maximal mean free path in the billiard table (or, equivalently, twice the logarithm of the joint spectral radius of the set of matrices determining reflections in the billiard walls). M. Zennaro, On the most stable switching laws of linear switched systems. We deal with discrete-time linear switched systems of the form x(n+ 1) = Aσ(n)x(n), σ : N→ {1, 2, . . . ,m}. where x(0) ∈ Rk, the matrix Aσ(n)∈Rk×k belongs to a finite family F = {Ai}1≤i≤m and σ denotes the switching law. It is known that the most stable switching laws are associated to the so-called spectrum-minimizing products of the family F . Moreover, for a normalized family F of matrices (i.e., with lower spectral radius ρ̌(F ) = 1) that share an invariant cone K, all the most stable trajectories starting from an initial value x(0) in the interior of K lie on the boundary of an antiball of a so-called invariant Barabanov antinorm. Under suitable conditions, a canonical constructive procedure for Barabanov antinorms of polytope type has been proposed by Guglielmi & Z. (2015). For families F sharing an invariant cone K, in this talk we show how to provide lower bounds to ρ̌(F ) by a suitable adaptation of the Gelfand limit in the framework of antinorms (Guglielmi & Z. (2019)), which could be of some practical interest when the above mentioned constructive procedure fails. Then we consider families of matrices F that share an invariant multicone Kmul (Brundu & Z. (2018, 2019)) and show how to generalize some of the known results on antinorms to this more general setting (Guglielmi & Z. (in progress)). These generalizations are of interest because common invariant multicones may well exist when common invariant cones do not F. Blanchini, Linear differential and difference inclusions: stability control and convergence. In this talk we consider the problems of stability analysis and stabilization of both linear difference and linear differential inclusions. In the first part a review of the history of the problem is proposed. In particular the equivalence will be described among the robust stability analysis, the spectral radius determination and the existence of a classes of Lyapunov functions. Duality properties and computational issues will be presented. It will be shown how these results have been extended to the robust stabilization problem. We finally present some recent results about: (a) structural stability analysis of biochemical networks (b) the problem of convergence in differential-difference inclusions. I. Morris, Fast approximation of the p-norm joint spectral radius. The joint spectral radius of a finite collection of matrices A1, . . . , AN describes the growth rate of the largest product of a given length which can be formed using those matrices. The p-norm joint spectral radius considers instead the growth rate of the `p-average of the norms of products of a given length, and has been studied on the basis of its applications to wavelet regularity and fractal geometry. I will describe an algorithm for approximating the p-norm joint spectral radius of positive matrices which is closely related to the work of Pollicott on approximating Lyapunov exponents.