Functional Analysis Approach to the Collatz Conjecture

Abstract

We examine the problems associated with the Collatz map T from the point of view of functional analysis. We associate with T a certain linear operator \(\mathcal {T}\) and show that cycles and (hypothetical) divergent trajectories (generated by T) correspond to certain classes of fixed points of the operator \(\mathcal {T}\). We also show the relationship between the dynamic properties of the operator \(\mathcal {T}\) and the map T. We prove that the absence of non-trivial cycles of T leads to hypercyclicity of the operator \(\mathcal {T}\). In the second part, we show that the index of the operator \(Id-\mathcal {T}\in \mathcal {L}(H^2(D))\) provides an upper estimate for the number of cycles of T. For the proof, we consider the adjoint operator \(\mathcal {F}=\mathcal {T}^*\)

$$\begin{aligned} \mathcal {F}: g\rightarrow g(z^2)+\frac{z^{-\frac{1}{3}}}{3}\left( g(z^{\frac{2}{3}})+e^{\frac{2\pi i}{3}}g(z^{\frac{2}{3}}e^{\frac{2\pi i}{3}})+e^{\frac{4\pi i}{3}}g(z^{\frac{2}{3}}e^{\frac{4\pi i}{3}})\right) , \end{aligned}$$

which was first introduced by Berg, Meinardus in [3], and show that it has no non-trivial fixed points in \(H^2(D)\). Furthermore, we calculate the resolvent of the operator \(\mathcal {F}\) and derive the equation for the characteristic function of the total stopping time \(\sigma _{\infty }\) as an application. In addition, 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.