# Fractal properties of the Hofstadter's butterfly and singular continuous spectrum of the critical almost Mathieu operator

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Abstract: Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly. It reduces to the critical almost Mathieu family, indexed by phase. We discuss the proof of sungular continuous spectrum for this family for all phases, finishing a program with a long history. We also discuss a recent proof (with I. Krasovsky) of the Thouless' conjecture from the early 80s: that Hausdorff dimension of the spectrum of Harper's operator is bounded by 1/2 for all irrational fluxes.