In the paper V. MÜLLER, A. PEPERKO, Generalized spectral radius and its max algebra version, LAA, 2013, 1006-1016 we (also) gave quite a simple proof of the Berger- Wang formula on the equality of the joint and the generalized spectral radius of a bounded set of non-negative (entrywise) n times n matrices, by first proving in a direct way its max algebra version.
The problem that remains here is the following : the reduction of the Berger Wang formula from the general case of a bounded set of (real)n times n matrices to the non-negative case. This would give a new simple-simpler (or at least an alternative) proof of the Berger Wang formula (in the n times n matrix setting). V. Muller noticed that this reduction is possible in the case of the singelton set {A} by using e.g. upper triangular Schur form of a matrix. What about e.g the case {A,B}?
This is a good question.
To get started, see the papers by V. Kozyakin on RG:
https://www.researchgate.net/profile/Victor_Kozyakin
For example, see
V. Kozyakin, Joint/Generalized Spectral Radius. Recent Advances, Russian Academy of Sciences, 2014
I just now found a paper that is more to the point for this question.
E. Garibaldi, J.T.A. Gomes, Aubry set asymptotically sub-additive potentials, 2014:
http://www.ime.unicamp.br/sites/default/files/rp13-14.pdf
The generalized spectral radius formula is given on page 8, This paper introduces a generalized Aubry set that makes it possible to show that the generalized spectral radius can be approximated by periodic matrix configurations. See Prop. 3.1, page 11.
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