Spectrum maximizing products are not generically unique

It is widely believed that typical finite families of $d\times d$ matrices admit finite products that attain the joint spectral radius. This conjecture is supported by computational experiments and it naturally leads to the following question: are these spectrum maximizing products typically unique, up to cyclic permutations and powers? We answer this question negatively. As discovered by Horowitz around fifty years ago, there are products of matrices that always have the same spectral radius despite not being cyclic permutations of one another. We show that the simplest Horowitz products can be spectrum maximizing in a robust way; more precisely, we exhibit a small but nonempty open subset of pairs of $2\times 2$ matrices $(A,B)$ for which the products $A^{2}BA{B}^2$ and $B^{2}AB{A}^2$ are both spectrum maximizing.