We introduce the notion of joint spectrum of a compact set of matrices $S\subset {GL}_d(\mathbb{C})$, which is a multi‐dimensional generalization of the joint spectral radius. We begin with a thorough study of its properties (under various assumptions: irreducibility, Zariski‐density, and domination). Several classical properties of the joint spectral radius are shown to hold in this generalized setting and an analogue of the Lagarias–Wang finiteness conjecture is discussed. Then we relate the joint spectrum to matrix valued random processes and study what points of it can be realized as Lyapunov vectors. We also show how the joint spectrum encodes all word metrics on reductive groups. Several examples are worked out in detail.

中文翻译：

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联合谱

我们介绍了一组紧凑矩阵的联合谱的概念$\mathrm{小号}\subset {GL}_d（\mathbb{C}）$，这是联合频谱半径的多维概括。我们从对其特性的全面研究开始（在各种假设下：不可约性，Zariski密度和支配性）。联合光谱半径的几个经典属性在这种广义背景下均成立，并讨论了Lagarias-Wang有限性猜想的一个类似物。然后，我们将联合频谱与矩阵值随机过程相关联，并研究可以将联合频谱的哪些点实现为Lyapunov向量。我们还将展示联合频谱如何对归约组上的所有单词度量进行编码。详细列举了几个例子。