We prove for the square Fibonacci Hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants. This is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact-dimensional measures.
David Damanik. Anton Gorodetski. Boris Solomyak. "Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian." Duke Math. J. 164 (8) 1603 - 1640, 1 June 2015. https://doi.org/10.1215/00127094-3119739