This article is cited in 5 scientific papers (total in 5 papers)
Self-affine polytopes. Applications to functional equations and matrix theory
A. S. Voynov
M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
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.
convex polytope, partition, functional equation, compression of the argument, stochastic matrix.
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Sbornik: Mathematics, 2011, 202:10, 1413–1439
MSC: Primary 52B12; Secondary 15B52, 37N99
A. S. Voynov, “Self-affine polytopes. Applications to functional equations and matrix theory”, Mat. Sb., 202:10 (2011), 3–30; Sb. Math., 202:10 (2011), 1413–1439
Citation in format AMSBIB
\paper Self-affine polytopes. Applications to~functional equations and matrix theory
\jour Mat. Sb.
\jour Sb. Math.
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This publication is cited in the following articles:
Proc. Steklov Inst. Math., 275 (2011), 290–292
Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
A. S. Voynov, “On the structure of self-affine convex bodies”, Sb. Math., 204:8 (2013), 1122–1130
A. S. Voynov, V. Yu. Protasov, “Compact noncontraction semigroups of affine operators”, Sb. Math., 206:7 (2015), 921–940
Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408
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