Victor S. Kozyakin

A Dynamical Systems Construction of a Counterexample to the Finiteness Conjecture

In 1995 J. C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse. In 2003 V. D. Blondel, J. Theys and A. A. Vladimirov proposed another proof of a counterexample to the Finiteness Conjecture which extensively exploited combinatorial properties of matrix products. In the paper, it is proposed one more proof of a counterexample of the Finiteness Conjecture fulfilled in a traditional manner of the theory of dynamical systems. It is presented description of the structure of trajectories with the maximal growing rate in terms of extremal norms and associated with them so-called extremal trajectories. The construction of the counterexample is based on a detailed analysis of properties of extremal norms of two-dimensional positive matrices in which the technique of the Gram symbols is essentially used. At last, notions and properties of the rotation number for discontinuous orientation preserving circle maps play significant role in the proof.

The Berger–Wang formula for the Markovian joint spectral radius☆

Abstract The Berger–Wang formula establishes equality between the joint and generalized spectral radii of a set of matrices. For matrix products whose multipliers are applied not arbitrarily but in accordance with some Markovian law, there are also known analogs of the joint and generalized spectral radii. However, the known proofs of the Berger–Wang formula hardly can be directly applied in the case of Markovian products of matrices since they essentially rely on the arbitrariness of appearance of different matrices in the related matrix products. Nevertheless, as has been shown by X. Dai [1] the Berger–Wang formula is valid for the case of Markovian analogs of the joint and the generalized spectral radii too, although the proof in this case heavily exploits the more involved techniques of multiplicative ergodic theory. In the paper we propose a matrix theory construction allowing to deduce the Markovian analog of the Berger–Wang formula from the classical Berger–Wang formula.

ITERATIVE BUILDING OF BARABANOV NORMS AND COMPUTATION OF THE JOINT SPECTRAL RADIUS FOR MATRIX SETS

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In [18, 21] the method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.

The inflation of attractors and their discretization: the autonomous case

The effects of perturbation or discretization on a maximal attractor of autonomous dynamical systems is now well understood, with the perturbed or discretized system having a nearby maximal attractor [5, 7, 10]. In contrast, for nonautonomous systems the very concept of a nonautonomous attractor itself is still undergoing intensive development and perturbation or discretization results have been obtained ∗This work was supported by the DFG Forschungschwerpunkt “Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme”. †The author was partially supported by the Russian Foundation for Basic Researches Grant 97–01–00692.

An explicit Lipschitz constant for the joint spectral radius

Abstract In 2002, Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.

Further results on stability of asynchronous discrete-time linear systems

Focuses on the stability problem of discrete-time asynchronous linear systems, which can be viewed as linear systems with time-varying delays. Within this context, a stronger version of the necessity part of the classical Chazan-Miranker theorem is proved and new results for special classes of system matrices are also presented.

Expansivity of semi-hyperbolic Lipschitz mappings

Semi-hyperbolic dynamical systems generated by Lipschitz mappings are shown to be exponentially expansive, locally at least, and explicit rates of expansion are determined. The result is applicable to nonsmooth noninvertible systems such as those with hysteresis effects as well as to classical systems involving hyperbolic diffeomorphisms.

Bi-shadowing and delay equations

Bi-shadowing is an extension to the concept of shadowing and is usually used in the context of comparing computed trajectories with the true trajectories of a dynamical system in Rn. Here the concept is defined in a Banach space and is applied to delay equations to give an apparently new result on nonlinear perturbations of linear delay equations. This is essentially a form of robustness with respect to small nonlinear disturbances.

On accuracy of approximation of the spectral radius by the Gelfand formula

Abstract The famous Gelfand formula ρ ( A ) = limsup n → ∞ ‖ A n ‖ 1 / n for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities ‖ A n ‖ 1 / n to ρ ( A ) . In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities ‖ A n ‖ 1 / n to ρ ( A ) . The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.

FURTHER RESULTS ON CONVERGENCE OF ASYNCHRONOUS LINEAR ITERATIONS

Abstract This paper focuses on the convergence problem of asynchronous linear iterations. A stronger version of the necessity part of the classical Chazan-Miranker theorem is proved and new results for special classes of iteration matrices are also presented.