A new class of $α$-Farey maps and an application to normal numbers
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929347171975168 |
|---|---|
| author | Dajani, Karma Kraaikamp, Cornelis Nakada, Hitoshi Natsui, Rie |
| author_facet | Dajani, Karma Kraaikamp, Cornelis Nakada, Hitoshi Natsui, Rie |
| contents | We define two types of the $α$-Farey maps $F_α$ and $F_{α, \flat}$ for $0 < α< \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le α\le 1$ by R.~Natsui (2004). Then, for each $0 < α< \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{α, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associted with $α$-continued fractions does not vary by the choice of $α$, $0 < α< 1$. This extends the result by C.~Kraaikamp and H.~Nakada (2000). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A new class of $α$-Farey maps and an application to normal numbers Dajani, Karma Kraaikamp, Cornelis Nakada, Hitoshi Natsui, Rie Dynamical Systems 11K50, 37A10, 11J70, 37A44 We define two types of the $α$-Farey maps $F_α$ and $F_{α, \flat}$ for $0 < α< \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le α\le 1$ by R.~Natsui (2004). Then, for each $0 < α< \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{α, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associted with $α$-continued fractions does not vary by the choice of $α$, $0 < α< 1$. This extends the result by C.~Kraaikamp and H.~Nakada (2000). |
| title | A new class of $α$-Farey maps and an application to normal numbers |
| topic | Dynamical Systems 11K50, 37A10, 11J70, 37A44 |
| url | https://arxiv.org/abs/2405.10921 |