A $p$-adic Approach To Piecewise Polynomial Dynamical Systems

TL;DR

This paper employs $p$-adic numbers to analyze polynomial dynamical systems, providing partial cycle classifications and new insights into the Collatz map, potentially advancing understanding of its long-standing conjecture.

Contribution

It introduces a $p$-adic framework for categorizing cycles in polynomial dynamical systems and offers new results related to the Collatz map's non-trivial cycles.

Findings

Partial classification of cycles using $p$-adic numbers

Results on non-trivial Collatz cycles

Implications for the Collatz conjecture

Abstract

Using $p$-adic numbers, we partially categorize the cycles of a sizable class of polynomial dynamical systems. In turn, we prove a few results related to the non-trivial cycles of the $\textit{Collatz map}$ $\text{Col} : \mathbb{Z}_+ \to \mathbb{Z}_+$ defined by $$\text{Col}(n) = \begin{cases} 3n + 1, & n \text{ is odd;} \\ n/2, & \text{otherwise.}\end{cases}$$ Proving the non-existence of non-trivial Collatz cycles would reduce the Collatz conjecture to whether the Collatz map can diverge to infinity.