Self-affine polytopes. Applications to functional equations and matrix theory

A special kind of functional equation with compression of the argument — the affine self-similarity equation — is studied. The earlier known one-dimensional self-similarity equations are generalized to the multidimensional case of functions of several variables. A criterion for the existence and uniqueness of an $L_p$-solution is established.
Description of such equations involves classification of finite-dimensional convex self-affine compact sets. In this work properties of such objects are thoroughly analysed; in particular, a counterexample to the well-known conjecture about the structure of such bodies, which was put forward in 1991, is given. Applications of the results obtained include some facts about the convergence of products of stochastic matrices; also, criteria for the convergence of some subdivision algorithms are suggested.
Bibliography: 39 titles.