Classification of wandering finite gaps in degree-$d>2$ invariant laminations: the higher-degree extension of Thurston's No Wandering Triangle theorem

For a complex polynomial of degree $d\ge 2$ with connected Julia set, external-ray angles on the circle are acted on by the $d$-tupling map $\sigma_d$, and Thurston's invariant laminations model the way rays land together. A finite gap (the convex hull of $k\ge 3$ angles forming an equivalence class) is wandering if it is neither (pre)periodic, nor (pre)critical, nor (pre)Fatou, so that its forward images are pairwise disjoint. Thurston proved that degree-2 laminations admit no wandering triangles (indeed no wandering gaps at all) and explicitly posed the extension to degrees $d\ge 3$. The extension fails in the naive sense -- wandering triangles exist combinatorially for every $d\ge 3$ and are realized by cubic polynomials -- so the surviving problem is a classification: for each degree $d>2$, determine exactly which wandering finite gaps ($k$-gons with $3\le k\le d$ by Kiwi's bound) can occur in $\sigma_d$-invariant laminations, with which combinatorial dynamics, and which of them are induced by degree-$d$ polynomials. Only partial necessary conditions (Kiwi; Blokh--Levin; Childers) and cubic-case constructions are known: no wandering $k$-gon with $k\ge 4$ has been constructed, none is excluded, and the extended Fatou--Shishikura inequality of Blokh--Childers--Levin--Oversteegen--Schleicher is known to be sharp only in degree 3.

Background

For a complex polynomial $P$ of degree $d\ge 2$, 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 its boundary; $P$ is semiconjugate on $\mathbb{C}\setminus K_P$ to $z\mapsto z^d$ near infinity (a semiconjugacy is a continuous surjection commuting with the maps), so external rays (images of radial lines under a Riemann map of the basin of infinity) land on $J_P$ and are parametrized by angles $\theta\in\mathbb{S}^1$, on which $P$ acts by the $d$-tupling map $\sigma_d(\theta)=d\theta\pmod 1$. Thurston introduced $\sigma_d$-invariant laminations as a combinatorial model of the sets of angles whose rays land at a common point: a geodesic lamination is a collection of chords of the closed unit disk (leaves) with pairwise disjoint interiors, and it is $\sigma_d$-invariant if leaves map forward to leaves (or points) under $\sigma_d$ and the structure is compatible with taking preimages. The closures of the components of the disk complementary to the union of leaves are called gaps; a finite gap is the convex hull in the disk of a finite set of $k\ge 3$ boundary angles (a $k$-gon). A chord whose two endpoints have the same $\sigma_d$-image is critical, and a gap is precritical if one of its iterates carries a critical chord on its boundary. A gap is wandering if it is neither (pre)periodic, nor (pre)critical, nor (pre)Fatou (a Fatou gap is one that eventually maps onto a periodic gap on which the first return map is a covering of degree $\ge 2$, modeling a periodic Fatou domain); equivalently, the convex hulls of the successive images of its vertex set are pairwise disjoint. When $J_P$ is locally connected (every point has arbitrarily small connected neighborhoods), the sets of external rays landing at a common point form the classes of an invariant lamination, and branch points of $J_P$ (points whose removal leaves $\ge 3$ components) of order $k$ correspond to gaps with $k$ vertices.

Thurston's No Wandering Triangle theorem states that $\sigma_2$-invariant laminations admit no wandering triangles; since the Blokh--Levin bound below, applied with $d=2$, leaves no room for wandering finite gaps at all (it gives $\sum_i(|G_i|-2)\le 0$, excluding gaps with three or more vertices), degree-2 invariant laminations have no wandering gaps, so branch points of locally connected quadratic Julia sets are preperiodic or precritical. This rigidity was instrumental in the pinched-disk (combinatorial) model of the Mandelbrot set (the set of parameters $c$ for which the Julia set of $z^2+c$ is connected), a model due to Douady, Hubbard and Thurston, and Thurston explicitly posed the problem of extending the theorem to higher degrees.

For $d\ge 3$ only partial information exists. Kiwi proved that a wandering non-precritical gap of a $\sigma_d$-invariant lamination is at most a $d$-gon. Blokh and Levin proved that for any wandering collection of gaps $G_i$ with pairwise distinct grand orbits (grand orbits are the equivalence classes under forward and backward iteration) one has $\sum_i(|G_i|-2)\le d-2$ (with refinements counting cycles of infinite gaps); in particular a degree-3 wandering gap is necessarily a triangle, and in degree $d$ a single wandering gap is at most a $d$-gon. Levin had earlier shown that unicritical invariant laminations (those generated by a single critical class, modeling polynomials with one critical point) admit no wandering polygons in any degree. Childers proved that a wandering $N$-gon forces at least $N-1$ recurrent critical leaves (critical chords whose forward orbits keep returning arbitrarily close to the chord) with pairwise disjoint infinite orbits sharing a common omega-limit set (the set of accumulation points of a forward orbit). On the existence side, Blokh and Oversteegen constructed, for every $d\ge 3$, uncountably many $\sigma_d$-invariant laminations with wandering triangles, and actual weakly hyperbolic cubic polynomials (a weak-expansion class introduced by Blokh and Oversteegen, with locally connected Julia sets) whose locally connected dendritic Julia sets (dendritic: homeomorphic to a dendrite, a locally connected continuum containing no simple closed curve) have wandering branch points (with an independent reproof by Buff, Canela and Roesch); Blokh gave necessary conditions for wandering triangles in cubic laminations. Blokh, Curry and Oversteegen made the cubic construction flexible (wandering triangles whose orbits are dense in every subarc, so-called condense orbits) and proved that critical portraits -- combinatorial parameter data given by $d-1$ pairwise unlinked critical chords -- that generate wandering-triangle laminations are uncountably dense in the space $\mathcal{A}_3$ of cubic critical portraits. Blokh, Childers, Levin, Oversteegen and Schleicher extended the Fatou--Shishikura inequality (the classical bound of at most $d-1$ non-repelling cycles, i.e. attracting, parabolic or irrationally neutral periodic orbits) to

$\widehat{N}+N_{\mathrm{irr}}+\chi+\sum_i(\operatorname{val}(Q_i)-2)\le d-1,$

where $\widehat{N}$ counts non-repelling cycles, $N_{\mathrm{irr}}$ counts repelling cycles at which no periodic external rays land, $\chi=1$ if a wandering collection of non-(pre)critical branch continua $Q_i$ exists (continua or points carrying the limit sets of at least three external rays, with pairwise disjoint forward orbits), and $\operatorname{val}(Q_i)$ is the number of rays with limit sets in $Q_i$; the inequality is sharp in degree 3 via the cubic wandering branch points, and the authors state they believe it sharp in general. The higher-degree classification program remains active: the posthumous Thurston et al. text on degree-$d$-invariant laminations and the Blokh--Oversteegen--Ptacek--Timorin laminational models study parameter-side structures (the latter without classifying wandering gaps), Bhattacharya, Blokh and Schleicher extended the full Thurston machinery (including absence of wandering polygons) to unicritical laminations of arbitrary degree, and the 2026 preprint of Blokh, Oversteegen and Timorin on root laminations (a q-lamination being a lamination saturated under the angle-equivalence classes it defines) describes itself as the initial step toward classifying all degree-$d$ invariant q-laminations, still citing Kiwi's bound as the state of the art on wandering gaps.

Problem Statement

Let $d\ge 3$ and let $\sigma_d:\mathbb{S}^1\to\mathbb{S}^1$, $\sigma_d(\theta)=d\theta\pmod 1$, be the $d$-tupling map of the unit circle. Recall that a ($\sigma_d$-invariant) lamination is a collection of chords of the closed unit disk, called leaves, with pairwise disjoint interiors, mapping forward to leaves (or points) under $\sigma_d$ and compatible with taking preimages; that the closures of the complementary components of the union of leaves are its gaps, a finite gap being the convex hull in the disk of a finite set of $k\ge 3$ boundary angles (a $k$-gon); that a chord whose two endpoints have the same $\sigma_d$-image is critical, and a gap is precritical if one of its iterates carries a critical chord on its boundary; and that a gap is wandering if it is neither (pre)periodic, nor (pre)critical, nor (pre)Fatou (a Fatou gap eventually maps onto a periodic gap on which the first return map is a covering of degree $\ge 2$), so that the convex hulls of the successive images of its vertex set are pairwise disjoint. Give a complete characterization of the wandering finite gaps of $\sigma_d$-invariant laminations, consisting of the following jointly posed parts.

(a) For each pair $(d,k)$ with $3\le k\le d$, decide whether a $\sigma_d$-invariant lamination can contain a wandering $k$-gon. Existence is known only for $k=3$ (all $d\ge 3$); no example with $k\ge 4$ is known and none is excluded beyond Kiwi's bound $k\le d$ and the Blokh--Levin bound $\sum_i(|G_i|-2)\le d-2$ for wandering collections with distinct grand orbits (grand orbits are the equivalence classes under forward and backward iteration).

(b) Characterize the admissible combinatorial dynamics of a wandering finite gap -- the possible itineraries (the sequences of its images relative to the critical chords), limit leaves and associated recurrent critical leaves (critical chords whose forward orbits keep returning arbitrarily close to the chord) -- and decide whether the known necessary conditions (Kiwi's $d$-gon bound, the Blokh--Levin sum bound, Childers' $N-1$ recurrent critical leaves) are sharp, i.e. matched by examples in every degree.

(c) Determine which wandering-gap laminations are realized by degree-$d$ polynomials -- meaning that the quotient map of $\mathbb{S}^1$ by the lamination's equivalence classes is semiconjugate (related by a continuous onto map commuting with the dynamics) to the polynomial restricted to its locally connected Julia set -- beyond the known cubic realizations; in particular, decide whether the extended Fatou--Shishikura inequality of Blokh--Childers--Levin--Oversteegen--Schleicher is sharp in every degree $d\ge 3$ (it is sharp in degree 3, and its authors state they believe it sharp in general).

The common objective of (a)-(c) is the higher-degree replacement of Thurston's No Wandering Triangle theorem for all $d\ge 3$ and all $3\le k\le d$: a theorem (or system of theorems) from which a specialist can read off exactly which wandering finite gaps occur, which constraints on their dynamics are sharp, and which are dynamically realized.

References

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