Sharpness of the extended Fatou–Shishikura inequality: saturation of the critical budget by wandering branch continua in all degrees

The classical Fatou–Shishikura inequality states that a complex polynomial of degree $d$ has at most $d-1$ non-repelling cycles. Blokh, Childers, Levin, Oversteegen and Schleicher (2016) proved an extended version: a chain of inequalities that also counts repelling cycles at which no periodic external rays land and wandering collections of eventual branch continua — continua of the Julia set whose pairwise-disjoint forward images eventually carry at least three external rays — against a budget of weakly recurrent and escaping critical points. In degree 3 the extended inequality is sharp in a strong sense: uncountably many cubic polynomials with locally connected dendrite Julia sets (a dendrite is a locally connected continuum containing no simple closed curve) carry a wandering branch point of valence 3 whose presence alone exhausts the entire budget of $d-1=2$ critical points, so that all inequalities of the paper's Theorem 1.2 hold with equality, and the authors state that they believe the inequality is sharp in general. The problem asks whether this saturation persists in every degree: for each $d\ge3$ — the open cases are $d\ge4$ — does there exist a degree-$d$ polynomial whose wandering eventual branch continua, together with the cycles of Julia components carrying them, consume the full critical budget $d-1$; or is the maximum $M(d)$ of this wandering-continua load strictly below $d-1$ for some degree, so that the wandering-continua part of the extended inequality can be improved? Whether $M(d)=d-1$ for $d\ge4$ is open.

Background

Let $P$ be a complex polynomial of degree $d\ge2$ and consider its iterates. The filled Julia set $K_P$ is the set of points with bounded forward orbits, and the Julia set $J_P=\partial K_P$ is the boundary between bounded and escaping dynamics. Near infinity $P$ is conformally conjugate to $z\mapsto z^d$, so external rays — the images of straight radial lines under this conjugacy — foliate the basin of infinity and accumulate on $J_P$; when several external rays land at the same point of $J_P$, that point sees several directions of escape, like a branch point of a tree. A periodic point is repelling if its multiplier has modulus greater than 1 and non-repelling otherwise (attracting, parabolic, or irrationally indifferent, the last type splitting into Cremer points and Siegel-disk centers). Critical points — the zeros of $P'$ — are the finite steering resource of polynomial dynamics: Fatou and Shishikura proved that a degree-$d$ polynomial has at most $d-1$ non-repelling cycles (the Fatou–Shishikura inequality), and Shishikura's quasiconformal surgery — a technique that grafts holomorphic pieces together along quasiconformal homeomorphisms — shows that this bound is sharp in every degree.

The source paper refines the inequality by counting more dynamical features on the left and fewer critical points on the right. A continuum (a nonempty compact connected set, possibly a point) $Q\subset J_P$ is wandering if all its forward images are pairwise disjoint; for a wandering continuum, the number of external rays with limit sets in $P^n(Q)$ is finite (by results of Kiwi and Blokh–Levin) and eventually constant along the forward orbit, and this stabilized number is the eventual valence $\mathrm{eval}_{J_P}(Q)$; $Q$ is an eventual branch continuum if $\mathrm{eval}_{J_P}(Q)\ge3$, the dynamical analogue of a branch point. On the critical side one counts weakly recurrent critical points — critical points in the filled Julia set $K_P$ whose orbits never land on repelling or parabolic points and are not eventually separated from themselves by any finite family of rational ray pairs, i.e. pairs of (pre)periodic external rays landing at a common point — together with escaping critical points, sorted by whether they lie in periodic or in wandering components of $J_P$. The Extended Fatou–Shishikura Inequality (Theorem 1.2 of the source paper) states, in a weak form,

$$\widetilde N+N_{\mathrm{irr}}+\chi+\sum_i(\mathrm{eval}_{J_P}(Q_i)-2)\le d-1,$$

where $\widetilde N$ counts non-repelling cycles, $N_{\mathrm{irr}}$ counts repelling cycles at which no periodic external rays land, the $Q_i$ form a wandering collection of non-(pre)critical eventual branch continua, and $\chi$ records whether it is nonempty; the full theorem is a pair of sharper inequalities bounding the same quantities by the numbers of weakly recurrent and escaping critical points, whose total is at most $d-1$.

Wandering branch continua are the topological shadow of a classical combinatorial question of Thurston. When $J_P$ is connected and locally connected, the equivalence relation "two external angles have rays landing at the same point" is compatible with the angle map $\sigma_d(\theta)=d\theta\bmod1$ and defines a $\sigma_d$-invariant lamination: a collection of chords (leaves) with disjoint convex hulls in the closed unit disk that maps into itself under $\sigma_d$, whose complementary components are called gaps. A point of $J_P$ at which $k\ge3$ rays land corresponds to a gap with $k$ vertices, and the gap is wandering if all its images are pairwise disjoint. Thurston proved that degree-2 laminations have no wandering triangles — a key ingredient of his combinatorial model of the Mandelbrot set — and posed the problem of extending this to higher degrees. As a non-existence statement that extension is false: Blokh and Oversteegen constructed uncountably many cubic laminations with wandering triangles, realized by cubic polynomials whose Julia sets are locally connected dendrites (locally connected continua containing no simple closed curve) with a wandering branch point. What survives is classification: Kiwi proved that a wandering gap of a degree-$d$ invariant lamination has at most $d$ vertices; Blokh–Levin proved $\sum_i(|G'_i|-2)+N'\le d-2$ for a wandering collection of finite gaps $G_i$, where $N'$ counts cycles of infinite gaps; Childers showed that each wandering gap $G$ carries $|G|-1$ recurrent critical chords with pairwise disjoint orbits; and Bhattacharya–Blokh–Schleicher recently proved that unicritical laminations (those modeling $z^d+c$) have no wandering polygons in any degree. The classification of wandering gaps for general degrees $d>2$ remains open.

In this landscape the source paper states one explicit open problem of its own: sharpness of the extended inequality in degrees beyond 3. In degree 2 wandering eventual branch continua cannot exist — for connected, locally connected Julia sets this is Thurston's No Wandering Triangle theorem, and in general it already follows from the extended inequality itself, since a single wandering eventual branch continuum would consume at least two of the one available unit of budget — so the extended inequality collapses to the classical one, which is sharp. In degree 3 the count is sharp in the strongest sense: by the cubic constructions of Blokh and Oversteegen (wandering triangles and wandering gaps for weakly hyperbolic cubic polynomials; see also Blokh–Curry–Oversteegen and Buff–Canela–Roesch) there exist — uncountably many — cubic polynomials with locally connected nonseparating dendrite Julia sets containing a wandering non-(pre)critical branch point $z$ of valence 3, so that $N_{co}=1$ (one cycle of Julia components carrying the wandering continuum) together with $\mathrm{eval}_{J_P}(Q)-2=1$ exhausts the entire budget $d-1=2$, both critical points being weakly recurrent, and both inequalities of Theorem 1.2 hold with equality. The authors add that they "believe it is sharp in general too". Note that literal equality in the summed bound is also attained classically in every degree by polynomials with $d-1$ attracting cycles; the non-trivial content of the belief — matching the degree-3 evidence they cite — is that the wandering continua themselves can saturate the critical budget, which is the problem stated here.

Problem Statement

Let $M(d)$ be the supremum, over all complex polynomials $P$ of degree $d\ge2$ and all wandering collections of eventual branch continua of $P$, of the total load of wandering eventual branch continua in the extended Fatou–Shishikura inequality of Blokh–Childers–Levin–Oversteegen–Schleicher:

$$M(d)=\sup_{P,\{Q_i\},\{Q^0_j\}}\left(N_{co}(P)+\sum_i(\mathrm{eval}_{J_P}(Q_i)-2)+\chi(m^0)+\sum_j(\mathrm{eval}_{J_P}(Q_j^0)-2)\right),$$

where the supremum ranges over polynomials $P$ of degree $d$ and over wandering collections $\{Q_i\}\cup\{Q^0_j\}$ of eventual branch continua: continua $Q\subset J_P$ (the Julia set of $P$, i.e. the boundary of the set of points with bounded forward orbits) such that the forward images of all members of the collection are pairwise disjoint, no forward image contains a critical point of $P$, and the eventual valence $\mathrm{eval}_{J_P}(Q)$ — the number of external rays (images of radial lines under the conformal conjugacy of $P$ to $z\mapsto z^d$ near infinity) with limit sets in $P^n(Q)$, a finite number that is eventually constant along the forward orbit — is at least 3. Within such a collection, $\{Q_i\}$ consists of those continua contained in (pre)periodic components of $J_P$, $\{Q^0_j\}$ of those that are themselves components of $J_P$, $N_{co}(P)$ counts the cycles of components of $J_P$ containing wandering eventual branch continua of the collection, and $\chi(m^0)$ is 1 if the second family is nonempty and 0 otherwise. The summed inequality of Theorem 1.2 of the cited paper gives $M(d)\le d-1$. Known values: $M(2)=0$ — already Theorem 1.2 itself rules out wandering eventual branch continua in degree 2, since any single one would consume at least two of the single available unit of budget (in the locally connected case this is also Thurston's No Wandering Triangle theorem, which excludes wandering gaps in degree-2 laminations) — and $M(3)=2=d-1$, by the Blokh–Oversteegen cubic polynomials with a wandering valence-3 branch point.

The problem is to determine $M(d)$ for every degree $d\ge3$; in particular: is $M(d)=d-1$ for every $d\ge3$, as the authors of the extended inequality believe ("we believe it is sharp in general too")? An affirmative answer must construct, for each degree $d\ge4$ (in any model class — dendrite Julia sets, i.e. Julia sets that are locally connected continua without simple closed curves, laminations realized by polynomials, or otherwise), polynomials whose wandering eventual branch continua, together with the cycles of Julia components carrying them, consume the entire critical budget $d-1$, with equality throughout the chain of Theorem 1.2 in the regime of the cubic examples, where no non-repelling Fatou or Cremer data competes for the budget. A negative answer must prove that $M(d)<d-1$ for some degree $d\ge3$, i.e. exhibit a universal improvement of the wandering-continua part of the extended inequality that is valid for all degree-$d$ polynomials. A complete determination of $M(d)$ for all $d\ge3$, whatever the values, also resolves the problem.

References

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