The Cui–Peng–Tan conjecture asserts that among post-critically finite rational maps, the only ones whose Julia sets contain a simply connected (full) wandering continuum are the flexible Lattès maps, for which such continua are line segments in bi-infinite or one-side-infinite geodesics of the flat metric. The non-simply-connected side is settled — those wandering continua exist exactly when the map has a Cantor multicurve and are eventually Jordan curves — and the flexible-Lattès exception is known to be sharp, but the classification on the simply connected side remains open; the conjecture is posed in the authors' Section 8.5 'Perspectives', item (2).

Background

A rational map is a holomorphic self-map of the Riemann sphere C^\hat{\mathbb{C}} given by a ratio of polynomials. Its dynamics splits the sphere into the Fatou set, where the iterates form a normal family (tame behavior), and the Julia set J(f)J(f), the closed complement where chaotic behavior concentrates. A rational map is post-critically finite (PCF) if the forward orbit of every critical point is finite; PCF maps are the combinatorial building blocks of rational dynamics, and the Julia set of a PCF map is automatically connected.

A continuum is a compact connected set with at least two points. A continuum K⊂J(f)K\subset J(f) is wandering if its forward images are pairwise disjoint, i.e. fn(K)∩fm(K)=∅f^n(K)\cap f^m(K)=\emptyset whenever n≠mn\neq m; it is called simply connected in the terminology of Cui–Peng–Tan (equivalently, full in the terminology of Cui–Gao) if the complement C^∖K\hat{\mathbb{C}}\setminus K has exactly one component. A wandering continuum is always full, in the terminology of Cui–Gao, if every forward iterate fn(K)f^n(K) is full. Sullivan's no-wandering-domains theorem excludes wandering Fatou domains for rational maps, but wandering continua inside Julia sets do exist, and classifying them is a central topological question in one-dimensional holomorphic dynamics.

Lattès maps are the rational maps that semi-conjugate an affine self-map z↦az+bz\mapsto az+b (with ∣a∣>1|a|>1) of a flat torus C/Λ\mathbb{C}/\Lambda to the sphere through a fixed branched covering of the torus ramified over four points; every Lattès map is PCF, has Julia set equal to the whole sphere, and carries a quotient Euclidean (flat) metric. Lattès maps divide into rigid ones and flexible ones, the latter forming continuous one-parameter families obtained by varying the translation parameter; the flexible Lattès maps are precisely those admitting nontrivial deformations. Cui–Gao proved that a Lattès map has an always-full wandering continuum exactly when it is flexible, the continuum being a line segment in a bi-infinite or one-side-infinite geodesic of the flat metric — the projection of a segment of a geodesic invariant under the affine torus map.

For general PCF maps, Cui–Peng–Tan built a decomposition theory. A multicurve is a finite collection of pairwise disjoint essential (non-nullhomotopic and non-peripheral) simple closed curves in C^\hat{\mathbb{C}} minus the postcritical set (the union of the forward orbits of the critical points); when such curves recur under pullback one can renormalize — restrict an iterate of the map to a domain that returns to itself, producing an embedded simpler dynamical system with its own 'small' Julia set. The paper introduces the Cantor multicurve (a multicurve whose successive pullbacks generate a strictly increasing number of curves in each homotopy class, the pullbacks nesting into a Cantor-set-like family of pairwise disjoint essential curves) and proves that a PCF rational map admits a non-simply connected wandering continuum if and only if it has a Cantor multicurve, in which case the wandering continua are eventually Jordan curves; the quotient dynamics on the decomposed pieces is encoded by a dendrite dynamical system (a dendrite is a locally connected continuum containing no simple closed curve), and a folding surgery produces examples with prescribed renormalizations. Thus the two known mechanisms producing wandering continua are Cantor multicurves (Jordan-curve type) and the flexible torus translation (full type), and the natural question is whether these exhaust all possibilities. Mating — the operation that glues the filled Julia sets of two polynomials of equal degree along their boundaries to produce a candidate rational map or branched covering — is the authors' proposed first test case, since matings are typically PCF and combine the dynamics of the two polynomials.

Problem Statement

Conjecture (Cui–Peng–Tan, 2016, Section 8.5 'Perspectives', item (2); the problem is also posed explicitly by Cui–Gao). A post-critically finite rational map admits a simply connected (full) wandering continuum in its Julia set if and only if it is a flexible Lattès map. The 'if' direction is the theorem of Cui–Gao. The open problem is the converse: prove that no post-critically finite rational map other than a flexible Lattès map has a simply connected wandering continuum — or exhibit such a map.

References

  1. Guizhen Cui, Wenjuan Peng, Lei Tan, "Renormalizations and wandering Jordan curves of rational maps", Communications in Mathematical Physics 344 (2016), no. 1, 67–115, DOI 10.1007/s00220-016-2623-x, arXiv:1403.5024, https://arxiv.org/abs/1403.5024 (source paper)
  2. Guizhen Cui, Yan Gao, "Wandering continua for rational maps", Discrete and Continuous Dynamical Systems 36 (2016), no. 3, 1321–1329, DOI 10.3934/dcds.2016.36.1321, arXiv:1404.7606, https://arxiv.org/abs/1404.7606
  3. Guizhen Cui, Fei Yang, Luxian Yang, "Decomposition of rational maps by stable multicurves", International Mathematics Research Notices 2025 (2025), no. 11, art. rnaf147, DOI 10.1093/imrn/rnaf147, arXiv:2309.03464, https://arxiv.org/abs/2309.03464
  4. Guizhen Cui, Yiran Wang, "Rational maps with constant Thurston pullback mapping", arXiv:2507.04215 (2025), https://arxiv.org/abs/2507.04215
  5. Yan Gao, Luxian Yang, Jinsong Zeng, "Rigidity of McMullen Julia sets", arXiv:2606.24520 (2026), https://arxiv.org/abs/2606.24520
  6. Guizhen Cui, Wenjuan Peng, "On the cycles of components of disconnected Julia sets", Mathematische Annalen 381 (2020), no. 1–2, 971–1003, DOI 10.1007/s00208-020-02073-7, arXiv:1904.04541, https://arxiv.org/abs/1904.04541
  7. Alexander Blokh, Doug Childers, Genadi Levin, Lex Oversteegen, Dierk Schleicher, "An extended Fatou–Shishikura inequality and wandering branch continua for polynomials", Advances in Mathematics 288 (2016) 1121–1174, DOI 10.1016/j.aim.2015.10.020, arXiv:1001.0953, https://arxiv.org/abs/1001.0953
  8. Yan Gao, Lele Xu, Luxian Yang, "Non-recurrent rational maps with disconnected Julia set", Nonlinearity 39 (2026), no. 4, 045004, DOI 10.1088/1361-6544/ae54f4, arXiv:2509.06645, https://arxiv.org/abs/2509.06645