We consider the spectrum of the Fibonacci Hamiltonian for small
values of the coupling constant. It is known that this set is a Cantor set
of zero Lebesgue measure. We show that as the value of the coupling constant
approaches zero, the thickness of this Cantor set tends to infinity, and,
consequently, its Hausdorff dimension tends to one. Moreover, the length of
every gap tends to zero linearly. Finally, for sufficiently small coupling,
the sum of the spectrum with itself is an interval. The last result provides
a rigorous explanation of a phenomenon for the Fibonacci square lattice
discovered numerically by Even-Dar Mandel and Lifshitz. The proof is based
on a detailed study of the dynamics of the so called trace map. This is a
joint work with David Damanik.