Dynamical spectrum via determinant-free linear algebra

Abstract

We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra and complex analysis) to rigorously estimate the eigenvalues of every matrix simultaneously without doing any calculations on the matrices themselves. As a corollary, we obtain mixing rates for every system at once, as well as symmetry properties of densities associated to the system; we also find the spectral properties of a sequence of related factor systems.