Fractal properties of the Hofstadter's butterfly and the singularly continuous spectrum of critical almost Mathieu operators

Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Thouless theory of the quantum Hall effect. The Harper operator spectra, as a function of magnetic flux, are organized into a remarkable self-similar structure: Hofstadter's butterfly. I will present recent results on the measure and dimension of the spectrum of this operator. The problem also reduces to a direct integral over the phase of critical almost Mathieu operators, and I will also discuss a solution to a 40+ year old problem - proof of the absence of a point spectrum for these operators, for all phases. The proof is based on simple harmonic analysis and a new Fourier type transform. I will also discuss recent advances in Thouless' "Catalan conjecture".