Interplay between topology and fractality of the hofstadter butterfly

We show that the tree structure underlying the Hofstadter butterfly fractal is a topological entity, solely determined by the Chern numbers of the butterflies. The topological quanta that label every butterfly in the butterfly graph are the band and the gap Cherns, the latter being the quanta of Hall conductivity. The mathematical framework to build the butterfly fractal consists of eight generators represented by $3\times 3$ unimodular matrices with integer coefficients. In the iterative process, a parent butterfly produces a sextuplet of butterflies, each attached to a tail that itself is made up of an infinity of butterflies of monotonically decreasing sizes. This eightfold way of building the butterfly fractal identifies butterfly with a tail as the building blocks of the butterfly graph.