联合谱半径、Sturmian测度和有限性猜想
Joint spectral radius, Sturmian measures, and the finiteness conjecture
摘要：
定义一对2 × 2实矩阵$(A_0,A_1)\in M_2(\mathbb{R})^2$的联合谱半径为$r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$，即这两个矩阵乘积的范数的最优增长率. Lagarias-Wang有限性猜想，断言$r(A_0,A_1)$总是某个长度为n的乘积$A_{i_1}...A_{i_n}$的谱半径的n次方根，已经被Bousch & Mairese反驳，随后的反例由Blondel，Theys&Vladimirov; Kozyakin; Hare，Morris，Sidorov & Theys提出。在这篇文章中，我们介绍了一种新的方法来产生有限反例，并使用它来展示一个开子集的$M_2(\mathbb{R})^2$的属性，每个成员$(A_0,A_1)$子集产生不可数许多形式$(A_0, tA_1)$的反例。我们的方法采用遍历理论，特别是分析Sturmian不变的措施，这种方法允许一个简短的证明，参数$t$和Sturmian参数$P(t)$之间的关系是一个魔鬼的楼梯。
英文摘要：
The joint spectral radius of a pair of 2x2 real matrices $(A_0,A_1)\in M_2(\mathbb{R})^2$ is defined to be $r(A_0,A_1)= \limsup_{n\to\infty} \max \{\|A_{i_1}...A_{i_n}\|^{1/n}: i_j\in\{0,1\}\}$, the optimal growth rate of the norm of products of these matrices. The Lagarias-Wang finiteness conjecture, asserting that $r(A_0,A_1)$ is always the nth root of the spectral radius of some length-n product $A_{i_1}...A_{i_n}$, has been refuted by Bousch & Mairesse, with subsequent counterexamples presented by Blondel, Theys & Vladimirov; Kozyakin; Hare, Morris, Sidorov & Theys. In this article we introduce a new approach to generating finiteness counterexamples, and use this to exhibit an open subset of $M_2(\mathbb{R})^2$ with the property that each member $(A_0,A_1)$ of the subset generates uncountably many counterexamples of the form $(A_0, tA_1)$. Our methods employ ergodic theory, in particular the analysis of Sturmian invariant measures; this approach allows a short proof that the relation between the parameter $t$ and the Sturmian parameter $P(t)$ is a devil's staircase.
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