Parity Sequences of the 3x+1 Map on the 2-adic Integers and Euclidean Embedding

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.

[1]  Kenneth G. Monks,et al.  The autoconjugacy of the 3x+1 function , 2004, Discret. Math..

[2]  Richard E. Crandall,et al.  On the $‘‘3x+1”$ problem , 1978 .

[3]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[4]  Lynn E. Garner.  On heights in the Collatz 3n + 1 problem , 1985, Discret. Math..

[5]  Daniel J. Bernstein,et al.  The 3x+1 conjugacy map , 1996 .

[6]  O. Rozier.  The 3x+1 problem: a lower bound hypothesis , 2015, 1510.01610.

[7]  J. Coquet,et al.  A summation formula related to the binary digits , 1983 .

[8]  K. Matthews,et al.  A generalization of Hasse's generalization of the Syracuse algorithm , 1984 .

[9]  D. Chistyakov.  Fractal geometry for images of continuous embeddings ofp-adic numbers and solenoids into Euclidean spaces , 1996 .

[10]  Dj Daniel Bernstein,et al.  A non-iterative 2-adic statement of the 3N+1 conjecture , 1994 .

[11]  Peter-Tobias Stoll,et al.  The 3x + 1 Conjugacy Map over a Sturmian Word , 2009 .

[12]  R. Terras,et al.  A stopping time problem on the positive integers , 1976 .

[13]  C. J. Everett.  Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2 , 1977 .

[14]  A. Cuoco.  Visualizing the p -adic integers , 1991 .

[15]  Jeffrey C. Lagarias,et al.  The 3x + 1 Problem and its Generalizations , 1985 .

[16]  A. F. Monna.  Sur une transformation simple des nombres P-adiques en nombres reels , 1952 .

[17]  J. Lagarias.  The set of rational cycles for the 3x+1 problem , 1990 .

[18]  Vladimir Anashin,et al.  Ergodic Transformations in the Space of p‐Adic Integers , 2006, math/0602083.

[19]  A. Khrennikov,et al.  Ergodicity criteria for non-expanding transformations of 2-adic spheres , 2012, 1205.0615.