An explicit counterexample to the Lagarias-Wang finiteness conjecture

作者：Hare Kevin G^*; Morris Ian D; Sidorov Nikita; Theys Jacques

来源：Advances in Mathematics, 2011, 226(6): 4667-4701.

DOI：10.1016/j.aim.2010.12.012

• Joint spectral radius
• Finiteness conjecture
• Sturmian sequence
• Balanced word
• Fibonacci word
• Infinite product

The joint spectral radius of a finite set of real d x d 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 d x d 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 2 x 2 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 set
A(alpha*) := {((1)(0) (1)(1)), alpha(*) ((1)(1) (0)(1))}
we give an explicit value of
alpha(*) similar or equal to 0.749326546330367557943961948091344672091327370236064317358024 ...
such that A(alpha*) does not satisfy the finiteness property.

• 出版日期2011-4-1