Functional analysis approach to the Collatz conjecture

TL;DR

This paper applies functional analysis to the Collatz conjecture by associating a linear operator with the Collatz map, exploring fixed points, cycles, and dynamical properties to gain insights into the conjecture.

Contribution

It introduces a novel operator-theoretic framework for analyzing the Collatz map, linking fixed points and cycles to properties of the associated linear operator.

Findings

Absence of nontrivial cycles implies hypercyclicity of the operator.

The index of the operator provides an upper bound on the number of cycles.

The adjoint operator has no non-trivial fixed points in the Hardy space.

Abstract

We investigate the problems related to the Collatz map $T$ from the point of view of functional analysis. We associate with $T$ certain linear operator $\mathcal{T}$ and show that cycles and (hypothetical) diverging trajectory (generated by $T$) correspond to certain classes of fixed points of operator $\mathcal{T}$. Furthermore, we demonstrate connection between dynamical properties of operator $\mathcal{T}$ and map $T$. We prove that absence of nontrivial cycles of $T$ leads to hypercyclicity of operator $\mathcal{T}$. In the second part we show that the index of operator $Id-\mathcal{T}\in\mathcal{L}(H^2(D))$ gives upper estimate on the number of cycles of $T$. For the proof we consider the adjoint operator $\mathcal{F}=\mathcal{T}^*$ \[ \mathcal{F}: g\to g(z^2)+\frac{z^{-\frac{1}{3}}}{3}\left(g(z^{\frac{2}{3}})+e^{\frac{2\pi i}{3}}g(z^{\frac{2}{3}}e^{\frac{2\pi i}{3}})+e^{\frac{4\pi i}{3}}g(z^{\frac{2}{3}}e^{\frac{4\pi i}{3}})\right), \] first introduced by Berg, Meinardus in \cite{BM1994}, and show it does not have non-trivial fixed points in $H^2(D)$. Moreover, we calculate resolvent of operator $\mathcal{F}$ and as an application deduce equation for the characteristic function of total stopping time $\sigma_{\infty}$. Furthermore, we construct an invariant measure for $\mathcal{T}$ in a slightly different setup, and investigate how the operator $\mathcal{T}$ acts on generalized arithmetic progressions.