A note on two Collatz evolution flows

TL;DR

This work introduces two Collatz-inspired evolution flows: a continuum Fourier-side PDE flow on $L^2(\mathbb T)$ driven by the generalized Collatz operator $\mathbf C$, and a discrete-time evolution governed by $C$ with discrete derivatives. It proves global well-posedness for the PDE flow with a growth bound $\|u(t)\|\le e^{\sqrt{2}\,t}\|u_0\|$, and shows how nontrivial periodic orbits and unbounded Collatz trajectories correspond to specific spectral features in the Fourier-coefficient dynamics. In the discrete setting, it defines a conserved energy-like quantity and derives sharp bounds for energy terms dependent on $\alpha$ and $\beta$, along with a precise description of the discrete time derivatives in terms of residue classes modulo powers of two. The results reveal a tight connection between discrete Collatz dynamics and Fourier-analytic PDE techniques, offering a new lens to study cycle formation and growth behavior via spectral data and energy identities. Overall, the paper provides a structured framework linking Collatz iterations to continuum flow dynamics and discrete energy conservation principles, with potential implications for understanding long-time behavior in Collatz-type systems.

Abstract

Two evolution models based on the generalized Collatz operator are introduced. These models are characterized by coefficients $α$ and $β$ in the Collatz dynamics, and are suitably defined. Here, $α=β=1$, and $α=3$, $β=1$ 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 Szegő operator of Gérard 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.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.2
• proof
• Theorem 2.4: Existence and uniqueness
• proof
• Theorem 2.5
• proof
• Lemma 2.6
• proof