On $q$-Analogs of the $3x+1$ Dynamical System

Abstract

The $3x+1$ Conjecture asserts that the $T$-orbit of every positive integer $x$ contains $1$, where $T$ maps $x$ to $x/2$ for $x$ even and to $(3x+1)/2$ for $x$ odd. Several authors have studied the analogous map, $T_q$, which maps $x\in F_2[q]$ to $x/q$ if $q$ divides $x$ and $((1+q)x+1)/q$ otherwise. In particular, they showed that the $T_q$-orbit of every polynomial contains $1$. This seems analogous to the $3x+1$ conjecture, but does not prove the conjecture itself, as the dynamical systems involved are not conjugate via any correspondence between polynomials and positive integers. In this paper, we show that $T_q$ actually is conjugate to $T$ if we extend their domains to the ring of formal power series $F_2[[q]]$ and the 2-adic integers $\mathbb{Z}_2$, respectively. Thus, it is not polynomials that correspond to positive integers via conjugacy, but rather certain formal power series. We then generalize this result to the family of functions $T_{A,B}\colon F_2[[q]]\to F_2[[q]]$ mapping $x$ to $x/q$ if $q$ divides $x$ and $(Ax+B)/q$ otherwise, where $A,B\in F_2[[q]]$ are not divisible by $q$. Unlike $T_q$, some of these maps do have the property that polynomials correspond to the positive integers whose $T$-orbit contains $1$ via a conjugacy with $T$. We show that $T_{1,1+q^2}$ is one such map, and has the additional nice property that the orbit of every polynomial enters either the unique $2$-cycle or one of the two fixed points. Finally, the power series that correspond to the natural numbers via these conjugacies can be represented as rational numbers with odd denominators by replacing $q$ with $2$ and interpreting the resulting formal series as a 2-adic integer. Finding a simple closed form for even one such correspondence could settle the conjecture itself, and we provide some data along these lines for both $T_{1,1+q^2}$ and $T_q$.