Self-similarity of Farey Staircases

2003/06/09 by Sebastian Grynberg, Sebastian P. Grynberg, Grynberg, Sebastian +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #11K55 #Complex Systems and Time Series Analysis #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Theoretical and Computational Physics #math-ph #math.MP #msc:11K55

paper · pdf · doi:10.48550/arxiv.math-ph/0306024

20 pages, Latex, 6 figures

arxiv created 2003/06/09 · openalex publication_date 2003/06/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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