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Finiteness property of a bounded set of matrices with uniformly subperipheral spectrum
 J. Commun. Technol. Electron
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On a Devil’s staircase associated to the joint spectral radii of a family of pairs of matrices
, 2013
"... The joint spectral radius of a finite set of real d×d matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain oneparameter family of pairs of matrices, this maxi ..."
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The joint spectral radius of a finite set of real d×d matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain oneparameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic ratio which depends continuously upon the parameter. In this note we answer some open questions from that paper by showing that the dependence of the ratio function upon the parameter takes the form of a Devil’s staircase. We show in particular that this Devil’s staircase attains every rational value strictly between 0 and 1 on some interval, and attains irrational values only in a set of Hausdorff dimension zero. This result generalises to include certain oneparameter families considered by other authors. We also give explicit formulas for the preimages of both rational and irrational numbers under the ratio function, thereby establishing a large family of pairs of matrices for which the joint spectral radius may be calculated exactly.
Approximating the joint spectral radius using a genetic algorithm framework
 in 18th IFAC World Congress IFAC WC2011
"... Abstract: The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate of products of matrices in the set. This quantity appears in many applications but is known to be difficult to approximate. Several approaches to approximate the joint spectral radius involve ..."
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Abstract: The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate of products of matrices in the set. This quantity appears in many applications but is known to be difficult to approximate. Several approaches to approximate the joint spectral radius involve the construction of long products of matrices, or the construction of an appropriate extremal matrix norm. In this article we present a brief overview of several recent approximation algorithms and introduce a genetic algorithm approximation method. This new method does not give any accuracy guarantees but is quite fast in comparison to other techniques. The performances of the different methods are compared and are illustrated on some benchmark examples. Our results show that, for large sets of matrices or matrices of large dimension, our genetic algorithm may provide better estimates or estimates for situations where these are simply too expensive to compute with other methods. As an illustration of this we compute in less than a minute a bound on the capacity of a code avoiding a given forbidden pattern that improves the bound currently reported in the literature.
MATHER SETS FOR SEQUENCES OF MATRICES AND APPLICATIONS TO THE STUDY OF JOINT SPECTRAL RADII
, 2011
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Extremal sequences of polynomial complexity
"... The joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partia ..."
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The joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2 p (2 p+1 −1) for which every extremal sequence has subword complexity at least 2−p2np.
NONCONFORMAL REPELLERS AND THE CONTINUITY OF PRESSURE FOR MATRIX COCYCLES
"... ABSTRACT. The pressure function P (A, s) plays a fundamental role in the calculation of the dimension of “typical ” selfaffine sets, whereA = (A1,..., Ak) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A ..."
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ABSTRACT. The pressure function P (A, s) plays a fundamental role in the calculation of the dimension of “typical ” selfaffine sets, whereA = (A1,..., Ak) is the family of linear mappings in the corresponding generating iterated function system. We prove that this function depends continuously on A. As a consequence, we show that the dimension of “typical ” selfaffine sets is a continuous function of the defining maps. This resolves a folklore open problem in the community of fractal geometry. Furthermore we extend the continuity result to more general subadditive pressure functions generated by the norm of matrix products or generalized singular value functions for matrix cocycles, and obtain applications on the continuity of equilibrium measures and the Lyapunov spectrum of matrix cocycles. 1.