Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian

Abstract

We consider the trace map associated with the Fibonacci Hamiltonian as a\ndiffeomorphism on the invariant surface associated with a given coupling\nconstant and prove that the non-wandering set of this map is hyperbolic if the\ncoupling is sufficiently small. As a consequence, for these values of the\ncoupling constant, the local and global Hausdorff dimension and the local and\nglobal box counting dimension of the spectrum of the Fibonacci Hamiltonian all\ncoincide and are smooth functions of the coupling constant.\n