Attractors of dual continued fractions

Given a Farey-type map F with full branches in the extended Hecke group Γm±, 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 al...

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Bibliographic Details
Published in:Journal of number theory 2022-11, Vol.240, p.50-73
Main Author: Panti, Giovanni
Format: Article
Language:English
Subjects:
Continued fractions
Hecke groups
Iterated function systems
Natural extension
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Summary:Given a Farey-type map F with full branches in the extended Hecke group Γm±, 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 Mm that simultaneously linearizes all maps with branches in Γm±, and show that the resulting dual linearized iterated function system satisfies the strong open set condition. We explicitly compute the Hölder exponent of every Mm, generalizing Salem's results for the Minkowski question mark function M3.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2021.12.011