The finiteness conjecture for the generalized spectral radius of a set of matrices

@article{Lagarias1995TheFC,
  title={The finiteness conjecture for the generalized spectral radius of a set of matrices},
  author={Jeffrey C. Lagarias and Yang Wang},
  journal={Linear Algebra and its Applications},
  year={1995},
  volume={214},
  pages={17-42}
}
The generalized spectral radius\g9(∑) of a set ∑ of n × n matrices is \g9(∑) = lim supk→∞\g9k(∑)1k, where \g9k(∑) = sup{ϱ(A1A2…Ak): each Ai ∈ ∑}. The joint spectral radius\g9(∑) is \g9(∑) = lim supk→∞\g9k(∑)1k, where \g9k(∑) = sup{∥A1 … Ak∥:each Ai ∈ ∑}. It is known that \g9(∑) = \g9(∑) holds for any finite set ∑ of n × n matrices. The finiteness conjecture asserts that for any finite set ∑ of real n × n matrices there exists a finite k such that \g9(∑) = \g9(∑) = \g9k(∑)1k. The normed… 
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TLDR
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Ergodic properties of matrix equilibrium states
  • Ian D. Morris
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2017
Given a finite irreducible set of real $d\times d$ matrices $A_{1},\ldots ,A_{M}$ and a real parameter $s>0$ , there exists a unique shift-invariant equilibrium state on $\{1,\ldots
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