Attractors of dual continued fractions

TL;DR

This paper investigates the properties of dual continued fractions related to Farey-type maps in Hecke groups, proving their domains are always tame and constructing linearizations that satisfy the strong open set condition.

Contribution

It introduces a homeomorphism that linearizes maps with branches in Gamma_m and demonstrates the dual linearized system satisfies the strong open set condition.

Findings

The domain of the dual map F_# always contains intervals.

Constructed homeomorphisms M_m linearize all maps with branches in Gamma_m.

Explicitly computed the Holder exponent for each M_m, extending Salem's results.

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 M_3.