Odometers, Backward continued fractions and counting rationals
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866916811809751040 |
|---|---|
| author | Iommi, Godofredo Ponce, Mario |
| author_facet | Iommi, Godofredo Ponce, Mario |
| contents | It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed, Newman's map has the property that the orbit of zero provides the required bijection. Claudio Bonanno and Stefano Isola, using continued fractions expansions, described the dynamics of its first return time map T. They proved that it is topologically conjugated to the dyadic odometer. In this article, we prove that the correct numerical system needed to analyze this map is the backward continued fractions. Indeed, this approach has the advantage that it provides explicitly the action of T on the expansion. As a by-product, we naturally obtain an explicit formula for Minkowski's question mark function in terms of backward continued fractions. The whole point of Newman was to provide an explicit bijection, our approach shares the same taste for the explicit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08329 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Odometers, Backward continued fractions and counting rationals Iommi, Godofredo Ponce, Mario Dynamical Systems Number Theory It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed, Newman's map has the property that the orbit of zero provides the required bijection. Claudio Bonanno and Stefano Isola, using continued fractions expansions, described the dynamics of its first return time map T. They proved that it is topologically conjugated to the dyadic odometer. In this article, we prove that the correct numerical system needed to analyze this map is the backward continued fractions. Indeed, this approach has the advantage that it provides explicitly the action of T on the expansion. As a by-product, we naturally obtain an explicit formula for Minkowski's question mark function in terms of backward continued fractions. The whole point of Newman was to provide an explicit bijection, our approach shares the same taste for the explicit. |
| title | Odometers, Backward continued fractions and counting rationals |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2310.08329 |