Self-similarity of Farey Staircases

Details Self-similarity of Farey Staircases Self-similarity of Farey Staircases

2003-06-09

arxiv.org/abs/math-ph/0306024v1

Physics

Mathematical Physics

20 pages, Latex, 6 figures

Scientific paper

We study Cantor Staircases in physics that have the Farey-Brocot arrangement for the Q/P rational heights of stability intervals I(Q/P), and such that the length of I(Q/P)is a convex function of 1/P. Circle map staircases and the magnetization function fall in this category. We show that the fractal sets Ommega underlying these staircases are connected with key sets in Number Theory via their (alpha,f(alpha)) multifractal decomposition spectra. It follows that such sets Ommega are self similar when the usual (Euclidean) measure is replaced by the hyperbolic measure induced by the Farey-Brocot partition.

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