General Dynamics and Generation Mapping for Collatz-type Sequences

Abstract

Let an odd integer \(\mathcal{X}\) be expressed as $\left\{\sum\limits_{M > m}b_M2^M\right\}+2^m-1,$ where $b_M\in\{0,1\}$ and $2^m-1$ is referred to as the Governor. In Collatz-type functions, a high index Governor is eventually reduced to $2^1-1$. For the $3\mathcal{Z}+1$ sequence, the Governor occurring in the Trivial cycle is $2^1-1$, while for the $5\mathcal{Z}+1$ sequence, the Trivial Governors are $2^2-1$ and $2^1-1$. Therefore, in these specific sequences, the Collatz function reduces the Governor $2^m - 1$ to the Trivial Governor $2^{\mathcal{T}} - 1$. Once this Trivial Governor is reached, it can evolve to a higher index Governor through interactions with other terms. This feature allows $\mathcal{X}$ to reappear in a Collatz-type sequence, since $2^m - 1 = 2^{m - 1} + \cdots + 2^{\mathcal{T} + 1} + 2^{\mathcal{T}}+(2^{\mathcal{T}}-1).$ Thus, if $\mathcal{X}$ reappears, at least one odd ancestor of $\left\{\sum\limits_{M > m}b_M2^M\right\}+2^{m-1}+\cdots+2^{\mathcal{T}+1}+2^{\mathcal{T}}+(2^{\mathcal{T}}-1)$ must have the Governor $2^m-1$. Ancestor mapping shows that all odd ancestors of $\mathcal{X}$ have the Trivial Governor for the respective Collatz sequence. This implies that odd integers that repeat in the $3\mathcal{Z} + 1$ sequence have the Governor $2^1 - 1$, while those forming a repeating cycle in the $5\mathcal{Z} + 1$ sequence have either $2^2 - 1$ or $2^1 - 1$ as the Governor. Successor mapping for the $3\mathcal{Z} + 1$ sequence further indicates that there are no auxiliary cycles, as the Trivial Governor is always transformed into a different index Governor. Similarly, successor mapping for the $5\mathcal{Z} + 1$ sequence reveals that the smallest odd integers forming an auxiliary cycle are smaller than $2^5$. Finally, attempts to identify integers that diverge for the $3\mathcal{Z} + 1$ sequence suggest that no such integers exist.