The Spectrum of the Off-diagonal Fibonacci Operator
Doctor of Philosophy thesis
The family of off-diagonal Fibonacci operators can be considered as Jacobi matrices acting in .e2(Z) with diagonal entries zero and off-diagonal entries given by sequences in the hull of the Fibonacci substitution sequence. The spectrum is independent of the sequence chosen and thus the same for all operators in the family. The spectrum is purely singular continuous and has Lebesgue measure zero. We will consider the trace map and its relation to the spectrum. Upper and lower bounds for the Hausdorff and lower box counting dimensions of the spectrum can be found under certain restrictions of the elements of the Fibonacci substitution sequence, and results from hyperbolic dynamics can be used to show that equality can be achieved between the two dimensions.