Seminar Details
2007-03-27 (14.00) : "The equations of self-similarity and the joint spectral radius"
At Auditoire Euler
Organized by Mathematical Engineering
Speaker :
Protasov, Vladimir
Abstract :
The joint spectral characteristics of linear operators such as the joint spectral radius, p-radius, the Lyapunov exponent etc.
have found numerous applications.
The computing or estimating these characteristics are known to be hard. We present a new approach to this problem based
on the so-called self-similarity equation. Equations of this type naturally appear in the wavelets theory, approximation theory,
dynamical systems etc. We give the definition of general self-similarity equation and prove that for any family of affine operators this equation
has at most one L_p - solution. This solution exists precisely when the p-radius of this family is less then one, in which case the solution is unique and continuously depend on the
operators. Approximating recursively this solution we can estimate the p-radius, including the joint spectral radius (if p is infinite).
To this end we generalize the notion of Barabanov's norm for p-radii for any p.
As a corollary we derive a criterion for the L_p-contractibility property of linear operators and estimate the asymptotic growth of orbits for any point.
This, in particular, gives a geometric interpretation of the p-radius as a spectral radius of a certain infinite-dimensional operator.