Lagarias-Wang有限性猜想的一个显式反例
An explicit counterexample to the Lagarias-Wang finiteness conjecture
摘要：
有限集$d \times d$矩阵集的联合谱半径被定义为从该集合中提取的矩阵长积的最大可能指数增长率。一组矩阵被称为具有\emph{finiteness property}，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年证明了实$d \times d$矩阵的每一个有限集都满足有限性。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$2 \times 2$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\[ \mathsf{A}_{α_*}:= \{({cc}1&1\\0&1), α_*({cc}1&0\\1&1)\}\]，我们给出α_* \simeq 0.749326546330367557943961948091344672091327370236064317358024.]使得$\mathsf{A}_{α_*}$不满足有限性。
英文摘要：
The joint spectral radius of a finite set of real $d \times d$ 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 \emph{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 $d \times d$ 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 $2 \times 2$ 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 \[ \mathsf{A}_{α_*}:= \{({cc}1&1\\0&1), α_*({cc}1&0\\1&1)\}\] we give an explicit value of α_* \simeq 0.749326546330367557943961948091344672091327370236064317358024...] such that $\mathsf{A}_{α_*}$ does not satisfy the finiteness property.
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