Functional analysis approach to the Collatz conjecture

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.