An explicit counterexample to the Lagarias-Wang finiteness conjecture
An explicit counterexample to the Lagarias-Wang finiteness conjecture
复制标题
DOI：
10.1016/j.aim.2010.12.012
复制
发表时间：
2010-06
期刊：
影响因子：
--
通讯作者：
K. Hare;I. Morris;N. Sidorov;J. Theys
中科院分区：
文献类型：
--
作者：
K. Hare;I. Morris;N. Sidorov;J. Theys
定义有限个真实的d×d矩阵集合的联合谱半径为从该集合中抽取的矩阵的长积的最大可能指数增长率。一组矩阵被称为具有有限性，如果存在一个周期积达到这个最大增长率。J.C. Lagarias和Y. Wang在1995年证明了每个有限的真实的d×d矩阵集都满足有限性。然而，T. Bousch和J. Mairesse在2002年证明了有限性猜想的反例存在，特别是证明了存在一族包含反例的2×2矩阵对。随后，V.D. Blondel，J. Theys和A.A.弗拉基米罗夫和V.S. Kozyakin，但没有明确的反例的有限性猜想迄今已给出。本文的目的是解决这个问题，给出了第一个完全明确的描述的一个反例的Lagarias-Wang有限性猜想。也就是说，对于集合，我们给出一个显式的值，使得[公式：见正文]不满足有限性。
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 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×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 2×2 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 we give an explicit value of such that [Formula: see text] does not satisfy the finiteness property.