Fixed-point classification and orbit asymptotics for the Collatz operator $\mathcal{T}$: the open problems of Neklyudov's functional-analytic approach

The Collatz (3n+1) map $T$ can be encoded as a linear operator $\mathcal{T}$ on spaces of analytic functions on the unit disk by $\mathcal{T}(\sum_n c_n z^n)=\sum_n c_n z^{T(n)}$. Neklyudov showed that finite cycles and hypothetical divergent orbits of $T$ correspond to specific classes of fixed points of $\mathcal{T}$, and posed six open problems completing this operator-theoretic dictionary. One of them — chaoticity of $\mathcal{T}$ — was resolved affirmatively by Béhani (2023, published 2024), who proved $\mathcal{T}$ hypercyclic on weighted Bergman-space quotients unconditionally and, under a weight condition satisfied by the classical Bergman weight, chaotic, frequently hypercyclic, and ergodic there. The remaining five questions are open and form the problem stated here: (1) an intrinsic functional-analytic criterion separating polynomial fixed points (finite cycles), divergent-orbit fixed points, and the independently constructed analytic fixed points; (2) whether $\mathcal{T}$ has a nonzero fixed point analytic on the whole complex plane that is not a scalar multiple of the trivial-cycle fixed point $z+z^2$ — a negative answer would prove the absence of nontrivial Collatz cycles; (3) whether the fixed points tied to a hypothetical divergent trajectory linearly combine to a nonzero entire function; (4) the limiting behaviour of $\mathcal{T}^k z^\alpha$ over the full range of exponents $\alpha\in\mathbb{C}$, of which the non-integer instances are the essential cases; and (5) the critical $\lambda=1/2$ limit of iterates of weighted arithmetic-progression generating functions. Jointly these ask whether the fixed-point set and orbit asymptotics of $\mathcal{T}$ intrinsically encode the Collatz scenarios of cycles, divergence, or neither.

Background

The Collatz (or 3n+1) problem concerns the map sending a positive integer $n$ to $n/2$ if $n$ is even and $3n+1$ if $n$ is odd; the conjecture asserts that every orbit reaches 1. In the reduced (accelerated) form used here, $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$, so the conjecture becomes: every orbit enters the unique known cycle $(1,2)$. Neklyudov associates to $T$ a linear operator $\mathcal{T}$ on spaces of analytic functions on the open unit disk $D$, defined on power series by $\mathcal{T}(\sum_{n\ge0}c_nz^n)=\sum_{n\ge0}c_nz^{T(n)}$ — that is, $\mathcal{T}$ sends each monomial to the monomial indexed by the Collatz image. He studies $\mathcal{T}$ on the Bergman space $\mathcal{B}^2(D)$ (the Hilbert space of analytic functions on $D$ with square-summable Taylor coefficients, with norm $\|\sum c_n z^n\|^2=\sum |c_n|^2/(n+1)$) and on the quotient by the invariant subspace $\mathrm{span}\{1,z,z^2\}$, which removes the trivial cycle; the kernel of $\mathcal{T}$ is $\mathrm{span}\{z^{2k+1}-z^{6k+4}:k\ge0\}$. The dictionary between Collatz dynamics and fixed points is threefold. First, each finite cycle $(n_1,\dots,n_k)$ of $T$ yields the polynomial fixed point $\sum_i z^{n_i}$ of $\mathcal{T}$, and every polynomial fixed point is of this form up to the kernel (Lemma 1.1 of the source). Second, each hypothetical divergent orbit $\{T^k(m)\}_{k\ge1}$ yields an analytic fixed point $f_m(1,\cdot)+\sum_{k\ge1}z^{T^k(m)}$ (Lemma 2.4, extended to an infinite family in Remark 2.5). Third, Theorem 2.3 and Theorem 5.4 of the source produce further fixed points that exist unconditionally, independent of any cycle or divergent orbit: every $\lambda\in\sqrt2 D$ is an eigenvalue of infinite multiplicity of $\mathcal{T}$ on the Bergman quotient (the $\lambda=1$ eigenspace consisting of fixed points), and Theorem 5.4 constructs an explicit fixed point $FP_2\in A(D)$ different from the ones constructed in Theorem 2.3; Theorem 2.3 also yields, for each $\lambda\in\mathbb{C}\setminus\sqrt2 D$, an infinite family of eigenfunctions $h_m(\lambda,\cdot)\in A(D)$, which for $\lambda\neq1$ are not fixed points. The antecedent of this program is the work of Berg and Meinardus on the adjoint operator $\mathcal{F}=\mathcal{T}^*$ acting on $A(D)$, the space of all analytic functions on $D$ with the topology of uniform convergence on compact sets: in Neklyudov's formulation (Corollary 4.14), the Collatz conjecture is equivalent to the eigenspace of $\mathcal{F}$ for the eigenvalue 1 on $A(D)$ being exactly two-dimensional — so deciding uniqueness of fixed points in the disk-analytic category is as hard as the full Collatz conjecture. Dynamically, Neklyudov proved that if $T$ has no nontrivial cycles then $\mathcal{T}$ is hypercyclic on the Bergman quotient (an operator is hypercyclic if it has a vector with dense orbit, and chaotic if it is hypercyclic with a dense set of periodic points); Béhani later proved that $\mathcal{T}$ is hypercyclic on weighted Bergman quotients unconditionally, and that it is chaotic, frequently hypercyclic, and ergodic with respect to a Gaussian invariant measure of full support under a weight condition ($\sum_n 1/\omega(k\,2^n)<\infty$ for every $k\ge3$) satisfied by the classical Bergman weight. Two further constructions frame what remains. On monomials with arbitrary complex exponent, $\mathcal{T}(z^\alpha)=\frac{1+e^{\pi i\alpha}}{2}z^{\alpha/2}+\frac{1-e^{\pi i\alpha}}{2}z^{(3\alpha+1)/2}$, which for integer $\alpha$ reproduces $z^{T(\alpha)}$ but for non-integer $\alpha$ (e.g. rationals with odd denominator) no longer encodes any extension of the Collatz map. And for the generating functions $g^{\lambda,\beta}_{k,l}(z)=\sum_{m\ge0}\lambda^{\sigma_\infty(lm+k)}\beta^m z^{lm+k}$, where $\sigma_\infty(n)$ is the total stopping time (the number of steps for the $T$-orbit of $n$ to reach the trivial cycle, and $\infty$ if it never does) and $\beta\in(0,1]$ weights the progression $\{lm+k\}$, the iterates $\mathcal{T}^n g^{\lambda,\beta}_{0,1}$ converge to $0$ for $0<\beta<1$ and $0<\lambda<\tfrac12$, while for $\lambda=\beta=1$ the candidate limit fails to exist for many progressions (for instance $g^{1,1}_{0,1}$ produces a divergent iterate sequence).

Problem Statement

Let $T$ be the reduced Collatz map, $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$, and let $\mathcal{T}$ be the linear operator defined on analytic functions on the unit disk by $\mathcal{T}(\sum_{n\ge0}c_nz^n)=\sum_{n\ge0}c_nz^{T(n)}$, considered on the Bergman space $\mathcal{B}^2(D)$ and on its quotient by $\mathrm{span}\{1,z,z^2\}$, with $\mathrm{Ker}\,\mathcal{T}=\mathrm{span}\{z^{2k+1}-z^{6k+4}:k\ge0\}$. The problem is to complete the operator-theoretic dictionary between the dynamics of $T$ and the fixed points and orbits of $\mathcal{T}$ by settling the five questions left open in Neklyudov's paper (Results in Mathematics 79 (2024), article 140). Their common objective is a fixed-point and orbit theory of $\mathcal{T}$ with enough intrinsic structure that operator-theoretic properties alone — without reference to the Collatz dynamics — detect which Collatz scenario (finite cycle, divergent orbit, or neither) generates each fixed point, and how orbits behave where the monomial encoding breaks down. The five questions: (1) Fixed-point classification: find an intrinsic functional-analytic criterion that provably separates the three known classes of fixed points of $\mathcal{T}$: polynomial fixed points, which are exactly the finite sums $\sum_i z^{n_i}$ over cycles of $T$ up to $\mathrm{Ker}\,\mathcal{T}$; the fixed points $f_m(1,\cdot)+\sum_{k\ge1}z^{T^k(m)}$ associated with hypothetical divergent orbits; and the fixed points constructed in Theorems 2.3 and 5.4 that exist independently of any cycle or divergent orbit. (2) Entire fixed points: decide whether there exists a nonzero fixed point of $\mathcal{T}$ analytic on the whole complex plane that is not a scalar multiple of $z+z^2$, the fixed point of the trivial cycle $(1,2)$ (Remark 5.5 of the source). Every nontrivial cycle of $T$ yields the polynomial fixed point $\sum_i z^{n_i}$, which is entire and, by Lemma 1.1, not a scalar multiple of $z+z^2$, so a negative answer would prove that $T$ has no nontrivial cycles. (3) Divergent span: under the hypothesis that a divergent $T$-orbit exists, decide whether the linear space generated by the associated fixed points of Lemma 2.4 and Remark 2.5 contains a nonzero function analytic on the whole complex plane. (4) Non-integer exponents: determine the limiting behaviour of the iterates $\mathcal{T}^k z^\alpha$ as $k\to\infty$ over the full range of exponents $\alpha\in\mathbb{C}$, where the non-integer values (e.g. rationals with odd denominator) are the essential instances, and where $\mathcal{T}(z^\alpha)=\frac{1+e^{\pi i\alpha}}{2}z^{\alpha/2}+\frac{1-e^{\pi i\alpha}}{2}z^{(3\alpha+1)/2}$ and the monomial action no longer encodes any extension of the Collatz map. (5) Critical weight limit: compute $\lim_{n\to\infty}\mathcal{T}^n g^{\lambda,\beta}_{k,l}$ at the critical value $\lambda=1/2$ (with $\beta\in(0,1]$ as in Definition 4.6 of the source, and $k,l\in\mathbb{N}$), where $g^{\lambda,\beta}_{k,l}(z)=\sum_{m\ge0}\lambda^{\sigma_\infty(lm+k)}\beta^m z^{lm+k}$ and $\sigma_\infty$ is the total stopping time; the limit is known to be $0$ for $g^{\lambda,\beta}_{0,1}$ when $0<\beta<1$ and $0<\lambda<1/2$ (Lemma 5.1), and the candidate limit is known to fail for many progressions at $\lambda=\beta=1$, leaving the critical case $\lambda=1/2$ open. The chaoticity question originally posed alongside them (whether periodic points of $\mathcal{T}$ are dense) has been resolved affirmatively by Béhani and is not part of this problem.

References

1. M. Neklyudov, Functional Analysis Approach to the Collatz Conjecture, Results in Mathematics 79 (2024), article 140. DOI: 10.1007/s00025-024-02167-7, https://doi.org/10.1007/s00025-024-02167-7 (preprint arXiv:2106.11859, https://arxiv.org/abs/2106.11859)
2. V. Béhani, Linear dynamics of an operator associated to the Collatz map, Proceedings of the American Mathematical Society 152 (2024), 1191–1206. DOI: 10.1090/proc/16627, https://doi.org/10.1090/proc/16627 (preprint arXiv:2303.03203, https://arxiv.org/abs/2303.03203)
3. L. Berg, G. Meinardus, Functional equations connected with the Collatz problem, Results in Mathematics 25 (1994), 1–12. DOI: 10.1007/BF03323136, https://doi.org/10.1007/BF03323136
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