Irrational fjords and limb triviality in the parameter plane of antipode-preserving cubic maps

The family of antipode-preserving cubic rational maps $f_q(z)=z^2\frac{q-z}{1+\bar q z}$ is parametrized by $s=q^2$ in a disk compactified by a circle at infinity whose points $\infty_t$ are indexed by rotation numbers $t\in\mathbb{R}/\mathbb{Z}$ . Bonifant, Buff, and Milnor proved the Fjord Theorem: for every rational $t$ with odd denominator, the balanced parameter ray in the principal hyperbolic component $H_0$ lands at $\infty_t$ . Three interlocking questions from that paper remain open. (1) Limb triviality (their Conjecture 7.11): is the $t$ -limb $L_t=(\widehat{\mathbb{C}}_{\mathrm{circ}}\smallsetminus H_0)\cap W_t$ the single point $\{\infty_t\}$ for every rational $t$ with odd denominator? (2) Irrational fjords (their Remark 7.9): does the balanced ray land at $\infty_t$ for some irrational $t$ , with uncountably many such $t$ expected, given that only accumulation at $\infty_t$ is known for rotation numbers in certain dense open sets? (3) Parabolic accumulation (their Theorem 3.6(b)): are parameter rays of doubling-periodic angles that neither land at a parabolic point nor at the circle at infinity bounded away from the circle at infinity, accumulating instead on curves of parabolic parameters? None of the three questions has been settled in print.

Background

A rational map is a holomorphic self-map of the Riemann sphere $\widehat{\mathbb{C}}$ , given as a quotient of polynomials. Under iteration the sphere splits into the Fatou set, where the sequence of iterates is stable, and its complement, the Julia set. A rational map is hyperbolic when every critical point is attracted to a periodic attracting cycle; in a holomorphic family the hyperbolic parameters form an open set whose connected components are called hyperbolic components. The antipodal map $A(z)=-1/\bar z$ is the fixed-point-free involution of $\widehat{\mathbb{C}}$ carrying each point to the diametrically opposite point, and a rational map is antipode-preserving if $A\circ f=f\circ A$ . A classical theorem of Borsuk quoted in the source paper shows that a self-map of the sphere preserving antipodes must have odd degree, so degree three is the first nontrivial case.

Bonifant, Buff, and Milnor study the family

f_q(z)=z^2\frac{q-z}{1+\bar q z},\qquad q\in\mathbb{C},

the antipode-preserving cubic maps with a double zero at the origin, equivalently with superattracting fixed points $0$ and $\infty$ ; the two remaining free critical points form an antipodal pair. Since replacing $q$ by $-q$ gives a conjugate map, one works with the parameter $s=q^2$ , and the $s$ -plane is compactified to a closed disk (the circled plane, denoted $\widehat{\mathbb{C}}_{\mathrm{circ}}$ ) by a circle at infinity whose points are $\infty_t$ , $t\in\mathbb{R}/\mathbb{Z}$ : a curve with $|q|^2\to\infty$ tends to $\infty_t$ when its asymptotic rotation number tends to $t$ . The principal hyperbolic component $H_0$ is the component containing $q=0$ (where $f_0(z)=-z^3$ ); for parameters in $H_0$ both free critical points lie in the immediate basins (the components of the attracting basins containing the fixed points) of $0$ and $\infty$ , and the Julia set is a quasicircle, i.e. the image of a round circle under a bounded-distortion homeomorphism.

Near the superattracting fixed point $0$ there is a Bottcher coordinate, a conformal chart in which the map becomes the squaring map; its radial lines define dynamic internal rays in the basin of $0$ . Transporting internal angles to parameter space yields the parameter rays $R_\vartheta\subset H_0$ , labeled by angles $\vartheta\in\mathbb{R}/\mathbb{Z}$ that evolve under angle doubling in the $s$ -plane. The source paper develops a rotation theory for the doubling map on the circle: to each angle $\vartheta$ it associates a rotation number $t=t(\vartheta)\in\mathbb{R}/\mathbb{Z}$ , and for every $t$ that is irrational or rational with odd denominator it singles out a unique balanced angle $\vartheta_t$ in the associated rotation set (for $t=m/p$ with $p$ odd, the middle element of the $p$ -point periodic orbit; for irrational $t$ , the unique point splitting the invariant measure of the Cantor rotation set in half). The formal rotation number extends this invariant to a continuous map from the circled plane minus the origin onto the circle at infinity, and the wake $W_t$ is the compact connected set of parameters of formal rotation number $t$ ; these wakes meet only at the origin, and $W_t$ touches the circle at infinity exactly at $\infty_t$ . Writing $L=\widehat{\mathbb{C}}_{\mathrm{circ}}\smallsetminus H_0$ and $L_t=L\cap W_t$ , the $t$ -limbs $L_t$ are compact connected sets whose disjoint union is $L$ . A parabolic point is a periodic point whose multiplier is a root of unity. A fjord is the authors' name for a channel to a point $\infty_t$ of the circle at infinity (in the sense of access-path theory: a consistent choice of the component of $U\cap H_0$ for neighborhoods $U$ of $\infty_t$ ) determined by an internal ray that lands at that point.

The Fjord Theorem (Theorem 7.1 of the source) states that if $t$ is rational with odd denominator and $\vartheta_t$ is the balanced angle, then the parameter ray $R_{\vartheta_t}$ lands at $\infty_t$ . For a doubling-periodic rational angle $\vartheta$ , Theorem 3.6 of the same paper leaves three alternatives for the ray $R_\vartheta$ : (a) it lands at a parabolic point of $\partial H_0$ , (b) it accumulates on a curve of parabolic boundary points, or (c) it lands at a point $\infty_t$ of the circle at infinity. For the multicorns, the connectedness loci of the antiholomorphic unicritical polynomials $z\mapsto\bar z^d+c$ , the analogue of case (b) is a theorem: Inou and Mukherjee proved that the parameter rays accumulating on the boundaries of odd-period hyperbolic components (other than period one) do not land but accumulate on arcs of parabolic parameters. That proof rests on the antiholomorphic nature of the maps $z\mapsto\bar z^d+c$ (for instance, on the resulting real-analytic structure of the odd-period parabolic loci); the maps $f_q$ are holomorphic, the source paper cites the multicorn result only as an analogy, and the corresponding statements for this family are the subject of this problem.

Problem Statement

In the notation of the source paper (published numbering), let $H_0$ be the principal hyperbolic component of the family $f_q(z)=z^2\frac{q-z}{1+\bar q z}$ in the circled $s=q^2$ parameter plane $\widehat{\mathbb{C}}_{\mathrm{circ}}$ with circle at infinity $\{\infty_t:t\in\mathbb{R}/\mathbb{Z}\}$ , let $W_t$ be the wake of formal rotation number $t$ , let $L=\widehat{\mathbb{C}}_{\mathrm{circ}}\smallsetminus H_0$ and $L_t=L\cap W_t$ the $t$ -limb, and let $R_{\vartheta_t}$ be the parameter ray of the balanced angle $\vartheta_t$ . The common objective is a complete description of how the parameter rays of $H_0$ land and how the complement of $H_0$ meets the wakes at the circle at infinity, extending the Fjord Theorem beyond odd-denominator rational rotation numbers. The unresolved components are:

(1) Limb triviality (Conjecture 7.11). If $t$ is rational with odd denominator, does $L_t$ consist of the single point $\infty_t$ ? A positive answer would imply, by the argument given in the source, that for a comeager set of rotation numbers $t$ (a countable intersection of dense open subsets of $\mathbb{R}/\mathbb{Z}$ ) one has $L_t=\{\infty_t\}$ and the balanced ray $R_{\vartheta_t}$ lands at $\infty_t$ .

(2) Irrational fjords (Remark 7.9). Do there exist irrational $t\in\mathbb{R}/\mathbb{Z}$ such that the balanced ray $R_{\vartheta_t}\subset H_0$ lands at $\infty_t$ , i.e. lands through an irrational fjord? Uncountably many such $t$ are expected. What is proved (Lemma 7.10) is that for the rotation numbers in countably many dense open subsets of $\mathbb{R}/\mathbb{Z}$ the ray accumulates at $\infty_t$ ; the accumulation set is a compact connected subset of $W_t\cap\partial H_0$ that could contain more than one point, and landing is not known for any irrational $t$ .

(3) Parabolic accumulation (published Theorem 3.6(b), and the remark following it — Remark 3.6 in the arXiv preprint, where the trichotomy is Theorem 3.5 — in which the authors promise the boundedness for the sequel manuscript). If $\vartheta\in\mathbb{Q}/\mathbb{Z}$ is periodic under doubling and the parameter ray $R_\vartheta$ falls in case (b) of the trichotomy, i.e. it neither lands at a parabolic point of $\partial H_0$ nor at a point of the circle at infinity, is the ray bounded away from the circle at infinity, so that it accumulates on a curve of parabolic boundary points without landing? Numerical computations reported in the source support this; the proof was promised for the sequel manuscript and the statement still appears as a conjecture in the publicly available 2015 draft (its Remark 3.5).

A complete resolution settles (1), (2), and (3) as stated, without restricting the class of rotation numbers, angles, or landing behaviors beyond the hypotheses given; an affirmative answer to (1) settles (2) as a corollary by the comeager argument stated in (1), so (2) calls for a separate argument only if (1) is answered negatively.

References

Araceli Bonifant, Xavier Buff, John Milnor. Antipode preserving cubic maps: the Fjord theorem. Proc. London Math. Soc. (3) 116 (2018), no. 3, 670-728. DOI: 10.1112/plms.12075, https://doi.org/10.1112/plms.12075 (preprint arXiv:1512.01850, https://arxiv.org/abs/1512.01850)
Araceli Bonifant, Xavier Buff, John Milnor. On antipode preserving cubic maps. Unpublished manuscript, draft of February 15, 2015 (precursor of the announced sequel 'Antipode Preserving Cubic Maps II: Tongues and the Ring Locus', which is still listed as in preparation in Milnor's 2025 bibliography). https://www.math.stonybrook.edu/~jack/bbm.pdf
John Milnor. List of publications for John Willard Milnor (2025 version; lists 'Antipode Preserving Cubic Maps II: Tongues and the Ring Locus' among manuscripts in preparation). https://www.math.stonybrook.edu/~jack/milnor-pub-2025.pdf
Hiroyuki Inou, Sabyasachi Mukherjee. Non-landing parameter rays of the multicorns. Inventiones mathematicae 204 (2016), 869-893. DOI: 10.1007/s00222-015-0627-3, https://doi.org/10.1007/s00222-015-0627-3 (preprint arXiv:1406.3428, https://arxiv.org/abs/1406.3428)
Russell Lodge, Sabyasachi Mukherjee. Invisible tricorns in real slices of rational maps. Discrete and Continuous Dynamical Systems 41 (2021), 1755-1797. DOI: 10.3934/dcds.2020340, https://doi.org/10.3934/dcds.2020340 (preprint arXiv:1710.05071, https://arxiv.org/abs/1710.05071)
Melida Carranza Trejo, Monica Moreno Rocha. Puzzle Pieces, Bi-accessibility, and Connectivity of the Julia Set for Generalized Blaschke Products. arXiv:2604.16750 (2026), https://arxiv.org/abs/2604.16750
Hiroyuki Inou, Tomoki Kawahira. Accessible hyperbolic components in anti-holomorphic dynamics. arXiv:2203.12156 (2022), https://arxiv.org/abs/2203.12156
Thomas Sharland. Matings of cubic polynomials with a fixed critical point. Part II: alpha-symmetry of limbs. arXiv:2205.03506 (2022), https://arxiv.org/abs/2205.03506