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Mathematics > Dynamical Systems

arXiv:1805.00133 (math)
[Submitted on 30 Apr 2018 (v1), last revised 14 Jul 2019 (this version, v5)]

Title:Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding

Authors:Olivier Rozier
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Abstract: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.
Comments: 30 pages, 2 figures. Various improvements
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
Cite as: arXiv:1805.00133 [math.DS]
  (or arXiv:1805.00133v5 [math.DS] for this version)
  https://doi.org/10.48550/arXiv.1805.00133
Journal reference: Integers 19 (2019), article A8

Submission history

From: Olivier Rozier [view email]
[v1] Mon, 30 Apr 2018 23:42:42 UTC (2,078 KB)
[v2] Mon, 14 May 2018 08:08:15 UTC (2,079 KB)
[v3] Sun, 10 Jun 2018 21:57:21 UTC (2,079 KB)
[v4] Sun, 9 Dec 2018 19:39:38 UTC (2,105 KB)
[v5] Sun, 14 Jul 2019 17:03:11 UTC (2,105 KB)
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