Rigidity for extremal products of matrices: Sturmian sequences
Abstract: Let $A,B$ two $2\times2$ real matrices with determinant $1$. We will consider the following problem: given an integer $n \geq 1$, what can one say about the products of $A,B$ of length $n$ that maximize the spectral radius? It turns out that in certain situations, maximizing sequences are in some sense rigid, they are forced to follow a very particular pattern described by Sturmian sequences. I will present a geometric approach (joint work with Emmanuel Breuillard) as well as an ergodic-theoretic approach (joint work with Jairo Bochi) to analyse this and similar problems.