For a rational map, a wandering continuum is a non-degenerate continuum in the Julia set whose forward iterates are pairwise disjoint; it is always full when every iterate has connected complement. For post-critically finite rational maps, a dichotomy of Cui, Peng and Tan reduces wandering continua to two branches: continua that eventually become Jordan curves, governed by Cantor multicurves, and always full continua. Cui and Gao settled the Lattès branch: a Lattès map admits an always full wandering continuum exactly when it is flexible, the continuum then being a line segment in a torus geodesic. The open problem in full generality is to determine under what condition a post-critically finite rational map admits an always full wandering continuum — conjecturally only flexible Lattès maps do, so no map with hyperbolic orbifold does — and what geometric forms such continua can realize. The problem was posed as open by its authors in 2015/2016.

Background

Holomorphic dynamics studies the iteration of a rational map ff of degree at least 2 on the Riemann sphere C^\widehat{\mathbb{C}}. The sphere splits into the Fatou set, where the iterates form a normal family (tame behavior), and its complement, the Julia set JfJ_f, where the dynamics is chaotic. A continuum is a connected compact set containing more than one point. A continuum K⊂JfK\subset J_f is a wandering continuum if its forward images are pairwise disjoint, i.e. fn(K)∩fm(K)=∅f^n(K)\cap f^m(K)=\emptyset for all n>m≥0n>m\ge 0; it is full if C^∖K\widehat{\mathbb{C}}\setminus K is connected, and always full if fn(K)f^n(K) is full for every n≥0n\ge 0. Wandering continua are the connected-Julia-set analogue of wandering Julia components: for geometrically finite rational maps (maps whose critical points in the Julia set are eventually periodic or eventually land on parabolic cycles) with disconnected Julia set, each wandering Julia component is known to be a point or a Jordan curve, and for polynomials with connected Julia set (absent irrational indifferent cycles) the nonexistence of wandering continua is equivalent to local connectivity of the Julia set. For non-polynomial rational maps with connected Julia set the situation is genuinely different, and the natural test class is the post-critically finite maps.

A rational map is post-critically finite (PCF) if the postcritical set Pf=⋃n>0fn(Ωf)P_f=\bigcup_{n>0}f^n(\Omega_f), where Ωf\Omega_f is the set of critical points (points of local degree greater than 1), is finite. PCF maps are the standard combinatorial models for sub-hyperbolic dynamics — rational maps that are eventually expanding away from finitely many eventually periodic critical orbits — and their combinatorics is finite, which is what makes an if-and-only-if classification a realistic goal. To each PCF map one attaches an orbifold Of=(C^,νf)\mathcal{O}_f=(\widehat{\mathbb{C}},\nu_f) by weighting each postcritical point zz with the least common multiple, over all iterates fnf^n with n>0n>0, of the local degrees of fnf^n at the points mapped to zz by fnf^n; the signature is the list of these weights, and the orbifold Euler characteristic χ(Of)=2−∑z(1−1/νf(z))\chi(\mathcal{O}_f)=2-\sum_{z}(1-1/\nu_f(z)) is always at most 0. The orbifold is parabolic when χ(Of)=0\chi(\mathcal{O}_f)=0 (the possible signatures being (∞,∞)(\infty,\infty), (2,2,∞)(2,2,\infty), (2,2,2,2)(2,2,2,2), (3,3,3)(3,3,3), (2,4,4)(2,4,4), (2,3,6)(2,3,6)) and hyperbolic when χ(Of)<0\chi(\mathcal{O}_f)<0. A Lattès map is a PCF map with parabolic orbifold and no infinite weight; equivalently, it is the factor of an expanding affine map A(z)=az+bA(z)=az+b of a flat torus C/Λ\mathbb{C}/\Lambda by a finite cyclic quotient of the torus, so its dynamics descends from toral affine dynamics. A Lattès map is called flexible when its orbifold has signature (2,2,2,2)(2,2,2,2) and the derivative aa of the covering affine map is an integer; these are exactly the Lattès maps admitting nontrivial quasiconformal deformations, i.e. deformations by homeomorphisms of the sphere that distort angles in a uniformly bounded manner. A multicurve is a non-empty finite collection of pairwise disjoint Jordan curves in C^∖Pf\widehat{\mathbb{C}}\setminus P_f, no two homotopic there, each of them essential in the sense that every complementary component contains at least two postcritical points; a Cantor multicurve is a multicurve for which the number of curves in each homotopy class under consecutive pullbacks tends to infinity, and it is the obstruction producing wandering Jordan curves.

The framework result, proved by Cui, Peng and Tan and recorded as Theorem A of Cui and Gao, is a dichotomy: for a PCF map ff, every wandering continuum is either always full, or is eventually mapped to Jordan curves, and the latter alternative can occur exactly when ff has a Cantor multicurve; equivalently — in the companion paper's terminology a continuum is called simply connected exactly when it is full — a PCF map has a non-simply-connected wandering continuum if and only if it has a Cantor multicurve, and such a continuum is eventually a Jordan curve. Thus the existence question splits into a solved branch (wandering Jordan curves, governed by Cantor multicurves) and the always-full branch, which is the subject of this problem. Cui and Gao solved the always-full branch within the Lattès class: a Lattès map admits an always full wandering continuum if and only if it is flexible, in which case the continuum is a line segment in a bi-infinite or one-side-infinite geodesic under the flat metric of the covering torus; they also observed that the remaining parabolic-orbifold classes — power maps (signature (∞,∞)(\infty,\infty)) and Chebyshev maps (signature (2,2,∞)(2,2,\infty)) — admit no wandering continua at all. Everything is therefore settled for parabolic orbifolds, and nothing is known for PCF maps with hyperbolic orbifold.

Problem Statement

Determine under what condition a post-critically finite rational map ff admits an always full wandering continuum: a non-degenerate continuum K⊂JfK\subset J_f such that fn(K)∩fm(K)=∅f^n(K)\cap f^m(K)=\emptyset for all n>m≥0n>m\ge 0 and C^∖fn(K)\widehat{\mathbb{C}}\setminus f^n(K) is connected for every n≥0n\ge 0. The parabolic-orbifold cases are already settled — power maps and Chebyshev maps have no wandering continua at all, and a Lattès map admits an always full wandering continuum if and only if it is flexible, the continuum then being a line segment in a torus geodesic — so the open core is the hyperbolic-orbifold class: does any post-critically finite rational map with χ(Of)<0\chi(\mathcal{O}_f)<0 admit an always full wandering continuum, and if so, under what precise condition on its combinatorial data (critical orbits, multicurves, orbifold type) and with what geometry? A complete answer is an if-and-only-if criterion over all post-critically finite rational maps; the conjectured criterion is that the flexible Lattès maps are the only ones. The problem includes the geometric sub-question of whether an always full wandering continuum of a post-critically finite map can take any form other than a line segment in a torus geodesic. The question is to be answered at this generality: restricting the class of maps, the admissible continua, or the notion of fullness would answer a different, derived problem.

References

  1. Guizhen Cui, Yan Gao, "Wandering continua for rational maps", Discrete and Continuous Dynamical Systems 36(3), 1321–1329 (2016). DOI: 10.3934/dcds.2016.36.1321, https://doi.org/10.3934/dcds.2016.36.1321 (preprint arXiv:1404.7606, https://arxiv.org/abs/1404.7606)
  2. Guizhen Cui, Wenjuan Peng, Lei Tan, "Renormalizations and wandering Jordan curves of rational maps", Communications in Mathematical Physics 344(1), 67–115 (2016). DOI: 10.1007/s00220-016-2623-x, https://doi.org/10.1007/s00220-016-2623-x (preprint arXiv:1403.5024, https://arxiv.org/abs/1403.5024)
  3. Yan Gao, Jinsong Zeng, Suo Zhao, "A characterization of Sierpiński carpet rational maps", Discrete and Continuous Dynamical Systems 37(9), 5049–5063 (2017). DOI: 10.3934/dcds.2017218, https://doi.org/10.3934/dcds.2017218 (preprint arXiv:1511.09191, https://arxiv.org/abs/1511.09191)
  4. Guizhen Cui, Yan Gao, Jinsong Zeng, "Invariant graphs of rational maps", Advances in Mathematics 404, 108454 (2022). DOI: 10.1016/j.aim.2022.108454, https://doi.org/10.1016/j.aim.2022.108454 (preprint arXiv:1907.02870, https://arxiv.org/abs/1907.02870)
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