Anton Gorodetski

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as

Hyperbolicity of the trace map for the weakly coupled Fibonacci Hamiltonian

\lambda \to \infty, {\rm dim} (\sigma(H_\lambda)) \cdot {\rm log} \lambda

The Fibonacci Hamiltonian

converges to an explicit constant,

MINIMAL AND STRANGE ATTRACTORS

{\rm log}(1+\sqrt{2})\approx 0.88137

Non-hyperbolic ergodic measures with large support

. We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schrödinger dynamics generated by the Fibonacci Hamiltonian.

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

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. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for a sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to study the spectral measures and the transport exponents.