A note on two Collatz evolution flows

TL;DR

This paper introduces two generalized Collatz evolution models, analyzes their dynamical properties, and establishes existence, boundedness, and energy conservation results for these flows, enhancing understanding of Collatz dynamics.

Contribution

It presents two new Collatz-based evolution models, proves existence and periodicity results, and introduces a conserved discrete energy for the second model.

Findings

Proved local and global existence in $L^2(\mathbb T)$ for the first model.

Characterized nontrivial periodic and unbounded orbits via solutions of the continuous flow.

Established energy conservation and growth bounds for the discrete model.

Abstract

Two evolution models based on the generalized Collatz operator are introduced. These models are characterized by coefficients $\alpha$ and $\beta$ in the Collatz dynamics, and are suitably defined. Here, $\alpha=\beta=1$, and $\alpha=3$, $\beta=1$ correspond to the Nollatz and classical Collatz operators, respectively. In general, the first evolution model is a continuum, Fourier side based, motivated by the Cubic Szeg\H{o} operator of G\'erard and Grellier. The second evolution considers discrete time derivatives of the Collatz orbits. In this paper we describe the evolution of both models, with particular emphasis on dynamical properties. For the first one, it is proved local and global existence in the space $L^2(\mathbb T)$, and a one-to-one characterization of the existence of nontrivial periodic and unbounded orbits of the Collatz mapping in terms of particular set of solutions of this continuous Collatz flow. For the discrete part, a sort of discrete energy is introduced. This energy has the property of being conserved by the discrete flow. An estimate of each term in this energy is given, proving suitable growth bounds. Finally, the meaning of the discrete time derivative for the generalized Collatz orbits is discussed. It is proved that, except for the Nollatz and Collatz operators, the sum of coefficients related to this discrete time derivative is an increasing sequence in $n$ as the iteration parameter $n$ evolves.