Parity sequences of the \(3x+1\) map on the 2-adic integers and Euclidean embedding. (English) Zbl 1448.11060
Summary: 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.
MSC:
| 11B83 | Special sequences and polynomials |
| 11B75 | Other combinatorial number theory |
| 37A25 | Ergodicity, mixing, rates of mixing |
Keywords:
inverse transform; induced automorphism; dynamics on the 2-adic integers; ergodic transformationOnline Encyclopedia of Integer Sequences:
a(n) = (-1 - (-2)^(n-2)) mod 2^n.Function of natural numbers satisfying the properties a(2*n) = 2*a(n) and a(2*n+1) = -3 + 2*a(3*n+2).
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