Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Functional analysis approach to the Collatz conjecture   We show how the problems related to the Collatz map $T$ can be transferred to
the language 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 connect the characteristic function of
total stopping time $\sigma_{\infty}$ with properties of linear operator \[
\mathcal{F}: g\to
g(z^2)+\frac{z^{-\frac{1}{3}}}{3}\left(g(z^{\frac{2}{3}})+e^{\frac{2\pi
i}{3}}g(z^{\frac{2}{3}}e^{\frac{2\pi i}{3}})+e^{\frac{4\pi
i}{3}}g(z^{\frac{2}{3}}e^{\frac{4\pi i}{3}})\right). \] Violation of Collatz
conjecture implies existence of fixed point of operator $\mathcal{F}$ of the
form \[ \sum\limits_{m\in\mathbb{N},\sigma_{\infty}(m)<\infty} z^m\in A(D). \]
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.

32
The categories of this paper are: atom-ph
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph