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

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 $\text{Collatz map}$ $\text{Col}:{\mathbb{Z}}_{+}\to {\mathbb{Z}}_{+}$ defined by $$\text{Col}(n)=\{\begin{array}{ll}3n+1,& n\text{ is odd;}\\ n/2,& \text{otherwise.}\end{array}$$ Proving the non-existence of non-trivial Collatz cycles would reduce the Collatz conjecture to whether the Collatz map can diverge to infinity.

5 pages