Attractors of dual continued fractions

2021/05/27 by Panti, Giovanni
#11A55 #37F32 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2105.13237

Abstract

Given a Farey-type map F with full branches in the extended Hecke group Gamma_m, its dual F_# results from constructing the natural extension of F, letting time go backwards, and projecting. Although numerical simulations may suggest otherwise, we show that the domain of F_# is always tame, that is, it always contains intervals. As a main technical tool we construct, for every m=3,4,5,..., a homeomorphism M_m that simultaneously linearizes all maps with branches in Gamma_m, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicitly compute the Holder exponent of every M_m, generalizing Salem's results for the Minkowski question mark function M₃.

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• On the Markoff spectrum on the Hecke group of index six

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