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Source. Theorem 2.2, p. 4, of Mikhail Neklyudov, Functional analysis approach to the Collatz conjecture, arXiv:2106.11859v9 (2022), published in Results Math. 79 (2024), no. 4, Paper No. 140, in the edition identified on the source card.

Statement

$T$ is the reduced Collatz map on $\mathbb Z$ and $\mathcal T$ the operator with $\mathcal T(z^n)=z^{T(n)}$, as on the Lemma 1.1 page. In Section 2 (p. 3) $\mathcal T$ acts on the quotient $H^2_{ber}(D)/X$ of the Bergman space of the unit disc, where $X=\operatorname{span}\{1,z,z^2\}$ is invariant, and has norm at most $2$ there.

Theorem 2.2 (p. 4). Quoted: "If $T$ has no nontrivial cycles than [sic] $\mathcal T$ is hypercyclic."

Hypercyclic means that some vector has a dense orbit under $\mathcal T$. The operator acts on power series, so only the values of $T$ on the nonnegative integers enter, and the cycles meant are cycles in $\mathbb N$; on the negative integers $T$ has other cycles, such as the fixed point $-1$.

Read depth. Claims checked: the statement was read clause by clause on p. 4; the proof was read through, not checked step by step.

Proof pointer

Page 4. The proof writes out only the case in which the $3n+1$ conjecture holds and says that the case of a diverging trajectory is similar. It applies the Godefroy--Shapiro hypercyclicity criterion (their Corollary 1.5, p. 235) with the right inverse $Sg(z)=g(z^2)$. The dense set on which iterates of $\mathcal T$ vanish comes from Lemma 2.1 (p. 3), which under the conjecture gives $H^2_{ber}(D)/X=\overline{\bigcup_{n\ge1}\operatorname{Ker}(\mathcal T^n)}$; the iterates of $S$ tend to zero on monomials.

Dependencies

Lemma 2.1 of the same paper (p. 3), which also proves $\mathcal T$ surjective on $H^2_{ber}(D)/X$.

Bears on

• #1135: a consequence of the absence of nontrivial cycles, which a positive answer to the problem would imply; it proves no case of the problem.