We consider solutions of a system of refinement equations written in the form φ = ∑ α∈Z a(α)φ(2 · −α), where the vector of functions φ = (φ1, . . . , φr)T is in (Lp(R)) and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R)) by Qaf := ∑ α∈Z a(α)f(2 · −α). This paper is concerned with the convergen...

We consider the smoothness of solutions of a system of reenement equations written in the form as = X 2ZZ a()(2 ?) where the vector of functions = (1 ; : : : ; r) T is in (L p (IR)) r and a is a nitely supported sequence of r r matrices called the reenement mask. We use the generalized Lipschitz space Lip (; L p (IR)), > 0, to measure smoothness of a given function. Our method is to relate the ...

We use ergodic theory to prove a quantitative version of a theorem of M. A. Berger and Y. Wang, which relates the joint spectral radius of a set of matrices to the spectral radii of finite products of those matrices. The proof rests on a theorem asserting the existence of a continuous invariant splitting for certain matrix cocycles defined over a minimal homeomorphism and having the property th...

Using multiplicative ergodic theory we prove two formulae describing the relationships between different joint spectral radii for sets of bounded linear operators acting on a Banach space. In particular we recover a formula recently proved by V. S. Shulman and Yu. V. Turovskĭı using operatortheoretic ideas. As a byproduct of our method we answer a question of J. E. Cohen on the limiting behavio...