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We address the problem of finite products that attain the joint spectral radius of a finite number of square matrices. Up to date the problem of existence of "forbidden products" remained open. We prove that the product AABABABB (together with its circular shifts and their mirror images) never delivers the strict maximum to the joint spectral radius if we restrict consideration to pairs \A, B\ of real 2 2 matrices. Under this restriction circular shifts and their mirror images constitute the class of isospectral products and hence they all have the same spectral radius for any pair \A, B\ of 2 2 matrices, even complex. For pairs of complex matrices we have numerical evidence that AABABABB is still a fobidden product. A couple of binary words that encode products from this isospectral class also happen to be the shortest forbidden patterns in the parametric family of double rotations.
А. А. Vladimirov (Tue,) studied this question.