The Julia set of a rational map always contains wandering points and even wandering Cantor sets, but every wandering continuum (a compact connected set with more than one point whose forward images are pairwise disjoint) currently known to exist in a rational Julia set is either an arc or a Jordan curve: McMullen's Cantor sets of wandering Jordan curves, the postcritically finite constructions of Cui–Peng–Tan, the wandering arcs near non-linearisable irrationally indifferent fixed points attributed to Cheraghi, and Cui–Gao's line-segment wandering continua for flexible Lattès maps. Successive rigidity results (Pilgrim–Tan for geometrically finite maps, point results for polynomials with disconnected Julia sets and for quadratic rational maps, the 2024 cubic bound of Cui–Peng–Yang on complementary components, and the 2025 non-recurrence result of Gao–Xu–Yang) progressively restrict the possible topology without excluding a continuum of any other type, while in transcendental dynamics wandering continua of essentially arbitrary topology are known to exist. The question, stated explicitly by Evdoridou–Martí-Pete–Rempe in 2026, is whether the Julia set of a rational map can contain a non-degenerate wandering continuum that is neither an arc nor a Jordan curve. No construction and no impossibility result is known: the problem is open.

Background

When a rational map f ⁣:C^→C^f\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}} of degree at least 2 is iterated, the Riemann sphere splits into the Fatou set — the open set on which the family of iterates {fn}\{f^{n}\} is normal, so nearby points share the same long-term behaviour — and its complement, the Julia set J(f)J(f), the locus of chaotic dynamics. Sullivan's celebrated no-wandering-domains theorem (1985) states that every Fatou component of a rational map is eventually periodic; the Julia set, by contrast, always contains wandering points, including points with dense orbits and a wandering Cantor set. A natural topological refinement asks which larger compact connected sets can wander inside the Julia set.

A continuum is a compact connected space with more than one point; an arc is a continuum homeomorphic to the closed interval [0,1][0,1], and a Jordan curve is one homeomorphic to the circle S1\mathbb{S}^{1}. A wandering continuum of a rational map ff is a continuum K⊆J(f)K\subseteq J(f) whose forward images are pairwise disjoint, fn(K)∩fm(K)=∅f^{n}(K)\cap f^{m}(K)=\emptyset for all n≠mn\neq m (the formulation of Cui–Gao). Exactly two topological types are known to occur. McMullen (1988) proved that for fλ(z)=z2+λ/z3f_{\lambda}(z)=z^{2}+\lambda/z^{3} with λ>0\lambda>0 sufficiently small the Julia set is a Cantor set of Jordan curves, uncountably many of which wander; Cui–Peng–Tan (2016) later constructed postcritically finite rational maps (maps whose postcritical set, the closure of the forward orbits of the critical points, is finite) with uncountably many wandering Jordan-curve Julia components, using Cantor multicurves and renormalization surgery, and showed that a wandering continuum of a postcritically finite map either is "always full" (its complement stays connected along the orbit) or eventually becomes a Jordan curve. Wandering arcs also occur: Cheraghi's analysis of quadratic polynomials with non-linearisable irrationally indifferent fixed points (also called Cremer points) — fixed points whose multiplier is e2πiαe^{2\pi i\alpha} with irrational α\alpha and which admit no holomorphic linearising change of coordinate — yields wandering arcs in their Julia sets, and Cui–Gao (2016) proved that a Lattès map, i.e. a rational map covered by an affine self-map of a flat torus, whose Julia set is the whole sphere, admits a wandering continuum exactly when it is flexible (its lifts include a nontrivial translation of the torus), in which case the continuum is a line segment under the flat metric, again an arc.

These are all the known examples, and several rigidity results show they are hard to extend. Pilgrim–Tan (2000) proved that for geometrically finite rational maps, i.e. those whose postcritical set accumulates only on attracting or parabolic cycles, every wandering component of the Julia set is a point or a Jordan curve. For polynomials, wandering Julia components of disconnected Julia sets are points, and a polynomial without irrationally indifferent cycles has no wandering continuum at all if and only if its Julia set is locally connected (results of Branner–Hubbard, Blokh–Levin, Kiwi, Levin, Qiu–Yin and Kozlovski–van Strien, surveyed in Cui–Gao). Wandering Julia components of quadratic rational maps are points. Most recently, Cui–Peng–Yang (2024) proved that every wandering Julia component of a cubic rational map eventually has at most two complementary components, and Gao–Xu–Yang (2025) proved that, under a non-recurrence hypothesis on the wandering components containing critical points, wandering "exposed" Julia components (a class of components singled out in that paper) are singletons. None of these constraints excludes a full wandering continuum with a branch point, such as a triod, or a non-Jordan continuum with two complementary components — not even for cubic maps.

The contrast with transcendental dynamics shows why the limit of the current techniques is a substantive question rather than a technicality. Rempe (2016) constructed a disjoint-type transcendental entire function — one whose singular values (critical and asymptotic values) all lie in the Fatou set — whose Julia set contains, for every arc-like continuum (an inverse limit of arcs; the class includes the sin⁡(1/x)\sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc), a wandering continuum homeomorphic to it, and Martí-Pete–Rempe–Waterman (2025) proved that a planar continuum is a Julia component of some transcendental meromorphic function precisely when it has empty interior, again via wandering continua. For rational maps nothing beyond arcs and Jordan curves is known, and the February 2026 preprint of Evdoridou–Martí-Pete–Rempe — the source of the question — records that, to the authors' knowledge, it is not known whether the Julia set of a rational map can contain a non-degenerate wandering continuum that is neither an arc nor a Jordan curve.

Problem Statement

Does there exist a rational map f ⁣:C^→C^f\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}} of degree at least 22 whose Julia set J(f)J(f) contains a non-degenerate wandering continuum that is neither an arc nor a Jordan curve? Concretely: is there a compact connected set K⊆J(f)K\subseteq J(f) with more than one point, with pairwise-disjoint forward images fn(K)∩fm(K)=∅f^{n}(K)\cap f^{m}(K)=\emptyset for all n≠mn\neq m, such that KK is homeomorphic neither to the closed interval [0,1][0,1] nor to the circle S1\mathbb{S}^{1} — for instance a continuum with a branch point such as a triod, a continuum whose complement has at least three connected components, or an indecomposable continuum (a continuum that cannot be written as the union of two of its proper subcontinua)?

References

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