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 Mat. Sb., 2011, Volume 202, Number 10, Pages 3–30 (Mi msb7791)

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

A. S. Voynov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

Keywords: convex polytope, partition, functional equation, compression of the argument, stochastic matrix.

DOI: https://doi.org/10.4213/sm7791

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English version:
Sbornik: Mathematics, 2011, 202:10, 1413–1439

Bibliographic databases:

UDC: 514.172.45+517.988.6+512.643.8
MSC: Primary 52B12; Secondary 15B52, 37N99

Citation: 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
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• http://mi.mathnet.ru/eng/msb7791
• https://doi.org/10.4213/sm7791
• http://mi.mathnet.ru/eng/msb/v202/i10/p3

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This publication is cited in the following articles:
1. Proc. Steklov Inst. Math., 275 (2011), 290–292
2. Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
3. A. S. Voynov, “On the structure of self-affine convex bodies”, Sb. Math., 204:8 (2013), 1122–1130
4. A. S. Voynov, V. Yu. Protasov, “Compact noncontraction semigroups of affine operators”, Sb. Math., 206:7 (2015), 921–940
5. Protasov V.Yu. Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408
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