کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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6426039 | 1345424 | 2011 | 35 صفحه PDF | دانلود رایگان |
The joint spectral radius of a finite set of real dÃd matrices is defined to be the maximum possible exponential rate of growth of long products of matrices drawn from that set. A set of matrices is said to have the finiteness property if there exists a periodic product which achieves this maximal rate of growth. J.C. Lagarias and Y. Wang conjectured in 1995 that every finite set of real dÃd matrices satisfies the finiteness property. However, T. Bousch and J. Mairesse proved in 2002 that counterexamples to the finiteness conjecture exist, showing in particular that there exists a family of pairs of 2Ã2 matrices which contains a counterexample. Similar results were subsequently given by V.D. Blondel, J. Theys and A.A. Vladimirov and by V.S. Kozyakin, but no explicit counterexample to the finiteness conjecture has so far been given. The purpose of this paper is to resolve this issue by giving the first completely explicit description of a counterexample to the Lagarias-Wang finiteness conjecture. Namely, for the setAαâ:={(1101),αâ(1011)} we give an explicit value ofαââ0.749326546330367557943961948091344672091327370236064317358024⦠such that Aαâ does not satisfy the finiteness property.
Journal: Advances in Mathematics - Volume 226, Issue 6, 1 April 2011, Pages 4667-4701