Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866912236403949568 |
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| author | Rozier, Olivier |
| author_facet | Rozier, Olivier |
| contents | In this paper, we consider the one-to-one correspondence between a 2-adic integer and its parity sequence under iteration of the so-called "3x+1" map. First, we prove a new formula for the inverse transform. Next, we briefly review what is known about the induced automorphism and study its dynamics on the 2-adic integers. We find that it is ergodic on many small odd invariant sets, and that it has two odd cycles of period 2 in addition to its two odd fixed points. Finally, a plane embedding is presented, for which we establish affine self-similarity by using functional equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_00133 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding Rozier, Olivier Dynamical Systems Number Theory In this paper, we consider the one-to-one correspondence between a 2-adic integer and its parity sequence under iteration of the so-called "3x+1" map. First, we prove a new formula for the inverse transform. Next, we briefly review what is known about the induced automorphism and study its dynamics on the 2-adic integers. We find that it is ergodic on many small odd invariant sets, and that it has two odd cycles of period 2 in addition to its two odd fixed points. Finally, a plane embedding is presented, for which we establish affine self-similarity by using functional equations. |
| title | Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/1805.00133 |