Noncommutative Perron-Frobenius-Ruelle theorem, two weight Hilbert   transform, and almost periodicity
  We consider the Jacobi matrix generated by a balanced measure of hyperbolic polynomial map. The conjecture of Bellissard says that this matrix should have an extremely strong periodicity property. We show how this conjecture is related to a certain noncommutative version of Bowen--Ruelle theory, and how the two weight Hilbert transform naturally appears in this context. 
A. Volberg and P. Yuditskii