Abstract

Co-authors: David Damanik (Rice University), William Yessen (Rice University)

In the talk we will consider the discrete Schrodinger operator with potential given by the Fibonacci substitution sequence (the Fibonacci Hamiltonian) and provide a detailed description of its spectrum and spectral characteristics (namely, the optimal Holder exponent of the integrated density of states, the dimension of the density of states measure, the dimension of the spectrum, and the upper transport exponent) for all values of the coupling constant. In particular, we will establish strict inequalities between the four spectral characteristics in question, and discuss the exact small and large coupling asymptotics of these spectral characteristics. A crucial ingredient is the relation between spectral properties of the Fibonacci Hamiltonian and dynamical properties of the Fibonacci trace map (such as dimensional characteristics of the non-wandering hyperbolic set and its measure of maximal entropy as well as other equilibrium measures, topological entropy, multipliers of periodic orbits). We will establish exact identities relating the spectral and dynamical quantities, and show the connection between the spectral quantities and the thermodynamic pressure function.