Rigidity of Newton dynamics
Abstract.
We study rigidity of rational maps that come from Newton’s root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filledin Julia set of a polynomiallike restriction of the original map. In particular, the Julia set is locally connected everywhere except possibly where it is renormalizable. As a corollary we show that for arbitrary Newton maps the boundary of every component of the basin of a root is locally connected.
In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either nonrenormalizable, or they are both renormalizable “in the same way”.
Our main tool is the concept of complex box mappings due to Kozlovski, Shen, van Strien; we also extend a dynamical rigidity result for such mappings so as to include irrationally indifferent or renormalizable situations.
2000 Mathematics Subject Classification:
30D05, 37F10, 37F201. Introduction and main results
1.1. Local connectivity, topological models, and rigidity
We investigate the fine structure in the dynamical systems formed by iteration of Newton maps of polynomials: the goal is to show that any two points within any given dynamical system can be distinguished in combinatorial terms (“dynamical rigidity”), and similarly that any two Newton dynamical systems can be distinguished combinatorially as well (“parameter rigidity”). Analogous rigidity results are known to be false for polynomial dynamics, and our main result is that they hold for the Newton dynamics everywhere except when embedded polynomial dynamics interferes (both in the dynamical plane and in parameter space). These results are strongest possible: embedded nonrigidity of polynomial dynamics makes rigidity in the Newton dynamics impossible.
This research connects to and builds upon a deep body of research on polynomial dynamics, initiated by Douady and Hubbard in their seminal Orsay Notes [DH1] and extended in celebrated work by Yoccoz [H], Lyubich and coauthors (see e.g. [L1, DL]), Kozlovski–van Strien [KvS1], and numerous others. The goal in much of this work is often phrased as showing that polynomial Julia sets are locally connected (many of these are, but not all; see for instance Milnor [M1]). The importance of local connectivity of Julia set comes from several closely connected aspects:

any two points in the Julia sets can be distinguished in terms of symbolic dynamics, for instance in the complement of pairs of dynamic rays that land at common periodic or preperiodic points.
For instance, Yoccoz’ theorem on quadratic polynomials can be phrased as saying that all quadratic polynomials that are nonrenormalizable and for which both fixed points are repelling have locally connected Julia sets, or equivalently that any two points in the Julia set can be distinguished in terms of their itineraries with respect to the single nondividing fixed point (usually called the fixed point).
Meanwhile, it is known that a polynomial Julia set is locally connected, and all its points can be distinguished in terms of symbolic dynamics, when the dynamics is not infinitely renormalizable and no periodic points are irrationally indifferent [KvS1]. In many cases with irrationally indifferent periodic points, or in the infinitely renormalizable setting, the Julia sets are locally connected anyway (compare e.g. [DR]); however, there are explicitly known examples when local connectivity fails, especially in the presence of Cremer points [M2, § 18] and in certain infinitely renormalizable cases [M1].
The research on local connectivity of polynomial Julia sets is among the deepest in all of dynamical systems. It has often been thought that the dynamics of rational maps must be even more complicated because polynomials have a basin of infinity that provides a simple and good coordinate system for the study of the dynamics, in particular through dynamic rays and their landing properties. In this paper, we propose a rather opposite point of view, at least for the dynamics of rational maps that are Newton maps of polynomials, that we phrase as the following principle:
Rational Rigidity Principle (dynamical version).
In the dynamics of any polynomial Newton map, the Julia set at any given point is locally connected, and the dynamics of can be distinguished by symbolic dynamics from any other point , unless the Newton dynamics is renormalizable and admits an embedded polynomial Julia set that fails to be locally connected or fails to be combinatorially rigid, and that contains the point (as well as ).
Here we say that a polynomial Julia set is embedded in the Newton dynamics when the latter is renormalizable and a domain of renormalization has a Julia set that is quasiconformally conjugate to the given polynomial Julia set.
There is a parallel discussion in parameter space that has also started with the work by Douady and Hubbard [DH1] on the Mandelbrot set:

if the Mandelbrot set is locally connected, then any two parameters in its boundary (the bifurcation locus) have Julia sets that can be distinguished in terms of symbolic dynamics;

if the Mandelbrot set is locally connected, then hyperbolic dynamics is open and dense in the space of quadratic polynomials.
For spaces of polynomial maps beyond quadratic polynomials, local connectivity is not the right concept (see a discussion below); instead the goal is to establish rigidity for instance in the form that any two polynomials for which the Julia sets are combinatorially indistinguishable are already quasiconformally conjugate. This rigidity conjecture is false in general, but it holds for instance when the polynomial dynamics is not renormalizable. Again, the study of parameter spaces of rational maps seems harder than for polynomials, but still we propose an analogous rigidity principle also in parameter space:
Rational Rigidity Principle (parameter space version).
Any two polynomial Newton maps that are combinatorially equivalent are quasiconformally equivalent, provided these Newton maps are either nonrenormalizable, or they are both renormalizable “in the same way”: the little Julia sets are hybrid equivalent and embedded into the Newton dynamics in combinatorially the same way.
Both versions of our rational rigidity principle, in the dynamical plane and in parameter space, can be interpreted as saying that “the Newton dynamics behaves well unless embedded polynomial dynamics interferes”, so contrary to frequent belief the dynamics of rational maps does not exhibit any additional complications beyond those known from polynomials, once a good combinatorial structure is established — at least in the case of polynomial Newton maps, which are the first family of rational maps for which a good combinatorial structure has been established [LMS1, LMS2].
1.2. Statement of results on rigidity
After this overview, we now provide a more precise statement of results. The Newton map of a polynomial is defined to be the rational map ; we call such a map a polynomial Newton map.
Our goal is to distinguish all orbits of in combinatorial terms; more precisely, in terms of symbolic dynamics. For polynomials, this is issue is closely related to the topology of the Julia set, in the sense that in many cases the distinction of all orbits is possible when the Julia set is locally connected. In analogy to [S1, S3], we define the fiber of a point as the set of points whose orbits are combinatorially indistinguishable from that of ; this is a compact connected set (see Definition 2.3 for a precise definition in general, and Section 4 specifically for Newton maps). We say that the fiber of is trivial if it consists of alone. Providing sufficient conditions for trivial fibers is one of the chief goals of this paper.
The purpose of Newton’s method is to find roots of : every root is an attracting fixed point of . The points that converge to these roots are known as the basins of the roots). They can be classified in terms of Newton graphs that connect the connected components of these basins to the roots; see [LMS1, LMS2]. From the dynamics point of view, the other points are more interesting: these points are either in the Julia set or are contained in Fatou components that eventually have period or higher (note that for every polynomial Newton map every Fatou component of period is a basin of a root because every fixed point is either attracting or the repelling fixed point at , and there cannot be Herman rings either [Sh]).
Our first main theorem (Theorem A) says that for every polynomial Newton map every point that is not attracted to a root can fail to have trivial fiber only if it belongs to (or is mapped to) an embedded quasiconformal copy of the filled Julia set of an actual polynomial mapping. This is the dynamical version of the Rational Rigidity Principle for Newton maps: rational Newton maps are dynamically rigid (have trivial fibers) except where polynomial dynamics interferes.
Theorem A (Dynamical Rigidity for Newton maps).
Let be a polynomial Newton map of degree . Then for every point at least one of the following possibilities holds true:

belongs to the Basin of attraction of a root of ;

has Trivial fiber;

belongs, or is mapped by some finite iterate, to the filled Julia set of Renormalizable dynamics (a polynomiallike restriction of with connected Julia set).
Theorem A immediately implies the following corollary.
Corollary 1.1 (Local connectivity or renormalizable dynamics).
Let be a polynomial Newton map and denote its Julia set by . Then for every , the set is locally connected at , except possibly when belongs to, or is mapped to, the Julia set of some renormalizable polynomiallike restriction of . ∎
Remark.
Many polynomial Julia sets are known to be locally connected, and even have all their fibers trivial, so the corresponding results can be imported to the Newton dynamics: if belongs to some renormalizable polynomiallike restriction of for which the polynomial Julia set has trivial fiber at the point corresponding to , then the Julia set of has trivial fiber at , and in particular is locally connected at . The idea of proof for this statement is that two points in a polynomial Julia set are in different fibers if and only if there is a pair of (pre)periodic dynamic rays landing at the same point in the polynomial Julia set that separates from , and within the dynamics of these rays can be replaced by “bubble rays” consisting of sequences of components of basins that converge to the same landing point with the same separation properties. (Note, however, that it is not clear that if a polynomial Julia set is locally connected at some point, then its fiber must be trivial, and this point can be separated from every other point in the Julia set; compare [S1, S3].)
There are polynomial Julia sets that are known not to be locally connected (for instance those having Cremer points, as well as some infinitely renormalizable polynomials). If such polynomial Julia sets happen to be ‘embedded’ in the Newton dynamical plane, the Julia set of the Newton map may or may not be locally connected at the corresponding points. Roesch [R] has shown that cubic Newton maps are locally connected in many cases even when they are renormalizable and the corresponding (quadratic) Julia sets are not.
Remark.
The degree of a Newton map is always equal to the number of distinct roots of (ignoring multiplicities). It is well known that if has degree , then the Julia set is a quasicircle through (in particular, it is a straight line if has degree as well), and the two complementary domains are the basins of the two roots. This case is trivial, and the case is even more trivial, so they are excluded from our discussions and we assume .
Another corollary of the dynamical rigidity of Newton maps (Theorem A) is the following result; a proof can be found in Section 5.
Corollary 1.2 (Boundary of the root basin locally connected).
For every Newton map, every component in the basin of any root has locally connected boundary.
Remark.
Note that in general the boundaries of the components of the basins of roots are not simple curves: they are pinched when the corresponding immediate basins have more than one access to . This may happen for all degrees .
Our second main result (Theorem B) is a parameter space counterpart to Theorem A. We say that two Newton maps are combinatorially equivalent if their Newton graphs coincide (see Definition 6.1 for details); an equivalent way of saying this is that the all the components of the basins of the roots are connected to each other in the same way (some examples are shown in Figure 1).
Theorem B (Parameter rigidity for Newton maps).
If two polynomial Newton maps are combinatorially equivalent, then they are quasiconformally conjugate in a neighborhood of the Julia set provided

either they are both nonrenormalizable,

or they are both renormalizable, and there is a bijection between their domains of renormalization that respects hybrid equivalence between the little Julia sets as well as their combinatorial position.
The domain of this quasiconformal conjugation can be chosen to include all Fatou components not in the basin of the roots, and its antiholomorphic derivative vanishes on those Fatou components as well as on the entire Julia set.
Moreover, if these Newton maps are normalized so that they attractingcriticallyfinite, then they are even affine conjugate.
The conditions in the renormalizable case mean the following: the renormalizable “little Julia sets” should correspond to the same polynomial dynamics (up to a quasiconformal conjugation that is conformal on the filledin Julia set of the polynomials), and they should be connected to the Newton graph (i.e. the graph consisting of all components of the basins of the roots) at the same combinatorial position. This will be made precise in Section 6.
Finally, a Newton map is called attractingcriticallyfinite if the orbit of every critical point in the basin of a root is eventually fixed; this can be accomplished by a routine quasiconformal surgery on a compact subset of the basins of the roots (see Section 4).
1.3. Box mappings
One of our key tools are complex box mappings as introduced by Kozlovski and van Strien [KvS1]. These maps are natural generalizations of polynomiallike maps to the case when the domains of definition and the ranges are disconnected. For any point in such a box mapping, the fiber is the set of points that have the same symbolic dynamics as with respect to the connected components for its domain of definition: that is, the set of points with the same itinerary through these connected components. Again, a precise definition will come later (Section 3). Kozlovski and van Strien show that certain box mappings have all their fibers trivial (in different language; see [KvS1, Theorem 1.4 (1)] and also [KvS2]). Our third main result (Theorem C) is an upgrade to their theorem: our result applies to all box mappings and provides sufficient conditions for most individual fibers to be trivial. Similarly as for polynomiallike maps, some points can only be iterated finitely many times; we say that such points escape (from the box mapping).
Theorem C (Generalized rigidity for complex box mappings).
Consider an arbitrary box mapping and an arbitrary nonescaping point . Then at least one of the following cases occurs:

has Trivial fiber;

belongs, or is mapped by some finite iterate, to the filled Julia set of Renormalizable dynamics (a polynomiallike restriction of the given box mapping with connected Julia set);

the domain of the box mapping contains a periodic component that maps onto itself by some iterate of the box map, and eventually maps to such a component (so the fiber of is equal to the closure of its component);

the orbit of Converges the Boundary of the domain of definition of the box mapping.
Observe that the first two possibilities here exactly match the two possibilities in Theorem A for points not in basins of the roots. The last two are “pathological” cases that are admitted by the fairly general definition of box mappings (see Definition 3.1), but do not naturally arise in many cases where box mappings are extracted from dynamical systems on (see also [KvS2]). This will be exactly the case in the proof of Theorem A. The letters (LABEL:Item:NE) denote an orbit that lands in a periodic component with No Escape: all points on this component remain there forever. A periodic component without escaping points will be called an (LABEL:Item:NE) component.
The concept of renormalization (in the sense of Douady–Hubbard, as well as of Kozlovski–van Strien) is discussed in Section 3.
Earlier work on Newton’s method. Newton’s method as a dynamical system has been studied by various people for a long time, in many cases with a focus on the cubic case: in this case there is a single free critical point and the parameter space is complex onedimensional, like the wellstudied case of the dynamics of quadratic polynomials and the Mandelbrot set. In particular, we would like to mention the classical work by Tan Lei [TL] with a combinatorial study of the Newton parameter space, with a recent refinement by Roesch, Wang, and Yin in [RWY]. In [R], Roesch has shown that the Newton map of a cubic polynomial has locally connected Julia set in many cases, even when it is renormalizable and the embedded polynomial Julia set is not locally connected.
More recently, there are two manuscripts that study Newton’s method of arbitrary degrees in a similar spirit as we do here. The main result of Wang, Yin, and Zeng [WYZ] is the result that immediate basins have locally connected boundary, similar to our Corollary 1.2. Roesch, Yin, and Zeng show in [RYZ] that all nonrenormalizable Newton maps are rigid (in parameter space); this corresponds to our Theorem B in the special case of nonrenormalizability.
There is also recent work on Newton’s method as an efficient root finder. Among the early results are a paper by Przytycki [Prz] that shows that immediate basins are always simply connected, and one by Manning [Ma] that shows where to start the Newton iteration to find some “exposed roots”. A sufficient small set of starting points that always finds all roots was constructed in [HSS] with a refinement in [BLS]. Estimates on the necessary number of iterations were given in [S2, S6, BAS]; see also the overview in [S4]. Finally, experiments that highlight the efficiency of Newton’s method for certain polynomials of degrees exceeding one billion were described in [SSt1, RSS].
Notation. In order to lighten notation, we write for the fold iterate of a map , that is .
We will also write for the set of critical points of a map , and put for the orbit of a point under the dynamics of .
2. On general puzzles
In this preparatory section we start with some general discussion and fix terminology concerning puzzles. In the end of the section we will prove triviality of fibers in some fairly general cases. These auxiliary results will be used later in the proof of the generalized rigidity principle — Theorem C.
Let be a holomorphic map between two open sets so that connected components of resp. have disjoint closures; we do not require that or be simply connected. We describe a setting of puzzles in the spirit of the well known Yoccoz puzzles, adapted to the needs of our Newton dynamics. Suppose that there exists a nested sequence of open sets such that , every component of is either compactly contained in or coincide with the corresponding component of and for every the restriction is a proper map. Further assume that the closure of each can be represented as a (not necessarily finite) union of closed topological disks ( runs over some finite or countable index set ) that can only intersect along their boundaries. We call each a puzzle piece of depth ; the union of all puzzle pieces of depth comprises the puzzle partition (of ) of depth . We will also call the topological graph the puzzle boundary of depth : vertices of this graph are either points on where at least two puzzle pieces meet, or points in where at least three puzzle pieces meet (note here that for every ); an edge of is a vertexfree set homeomorphic to an interval that connects two vertices. For simplicity we assume that all edges in all are smooth and the boundary of each puzzle piece contains finitely many vertices. (In the special case that is a component of such that for some , this definition of edges and vertices does not apply; in this case, we choose an arbitrary point on as a vertex and let the rest of be an edge that connects the vertex to itself.)
Remark.
The previous paragraph describes, in fairly large generality, a construction that includes not only the setting of the well known Yoccoz puzzles (where each consists of finitely many puzzle pieces), but it also caters for two settings that will be specified in the upcoming sections. First, we will be interested in puzzles for complex box mappings (see Definition 3.1). In that case, will be a box mapping, where each will be a (possibly infinite) union of open topological disks with disjoint closures; the closure of each of the disks will serve as a puzzle piece of depth . In other words, for a box mapping and for any given the set of puzzle pieces of depth equals the set of closures of the connected components of ; see Definition 3.2 for details. For the second time the construction in the previous paragraph will be specified for polynomial Newton maps (see Section 4). There will stand for a particularly chosen iterate of the Newton map, while will be the Riemann sphere minus finitely many suitably chosen closed topological disks bounded by equipotentials in the respective basins of roots of ; each of these removed disks is a neighborhood of either a root of or an iterated preimage of such a root for a bounded number of iterations (see Definition 4.3 for details).
Definition 2.1 (Markov property).
The union of all puzzle pieces of all depths has the Markov property if:

any two puzzle pieces are either nested or have disjoint interiors; in the former case the puzzle piece of bigger depth is contained in the puzzle piece of a smaller depth;

the image of each puzzle piece of depth is a puzzle piece of depth , and the restriction is a branched covering.
Equivalently, the Markov property can be stated in terms of puzzles: the union of all puzzle pieces of all depths has the Markov property if for all . (Note that for puzzles coming from box mappings this condition is automatically satisfied since , see Definition 3.2).
We will say that is a holomorphic map with welldefined Markov partition if , with , is a holomorphic map as described at the beginning of the section and for which there exists a nested sequence of open sets with a welldefined puzzle partition into puzzle pieces the union of which has the Markov property.
For a holomorphic map with a welldefined Markov partition, each puzzle piece of depth is a nice set in the sense of [L2, ]: the orbit of the boundary of does not intersect the interior of (that is for all possible ). This follows from the conditions and (both for all ).
Definition 2.2 (Puzzle piece centered at a point).
Given a point , define to be the union of all puzzle pieces of depth containing .
From the definition above it is clear that if is not on the boundary of any puzzle piece of depth (equivalently, if ), then is the unique puzzle piece of depth containing the point . Otherwise, is a union of puzzle pieces with in their common boundary. Note that these sets do not form a Markov partition: it may be that and are different with intersecting interiors if or are in . However, it is still true that the restriction is a branched covering.
Let us spell out an elementary argument that will be used several times below without explicit mention. If is a point that does not belong to the puzzle boundary of any depth, then every point on the orbit of also does not belong to the puzzle boundary of any depth, and hence is the puzzle piece (of depth ) for all and (as long as can be iterated times).
We say that a point escapes if for some . Thus the set of nonescaping points of (the nonescaping set of ) is precisely ; this is the set of points that can be iterated infinitely often.
Definition 2.3 (Fiber, trivial fiber).
For a nonescaping point , the set
is called the fiber of (with respect to the partition of ). We say that has trivial fiber if .
The Markov property of puzzle partitions is a powerful combinatorial property allowing us to study maps from the point of view of symbolic dynamics. We define the itinerary at level of a point as the sequence . Two points have the same itinerary if their itineraries at all levels coincide. In particular, the fiber consists of all points that have the same itineraries at all levels. In other words, consists of all points that are dynamically indistinguishable with respect to our puzzle partition. Hence, if the fiber is trivial, then the dynamics of at is rigid: there is no point other than with the same itinerary as .
We will also say that the fiber is periodic if has periodic itinerary; this property is independent of a particular choice of a point in the fiber.
A point is called combinatorially recurrent if does not belong to the puzzle boundary of any depth and the orbit of under intersects for every . This implies that the orbit of intersects every infinitely often: if this was not true, then for some there was a largest with , so for all ; hence . But then and are in the same fiber for every , so for all .
We start by a pair of fairly standard technical lemmas, and then proceed by proving some of the simplest cases when fibers are trivial.
Lemma 2.4 (Annulus pullback under branched covering).
Let be a branched covering of degree at most between two closed topological disks. Suppose is an open annulus with , and assume that an annulus is a component of . Then .
Proof.
The branched cover has at most critical points. Hence the annulus has a parallel subannulus of modulus that avoids all critical values (recall that is a parallel subannulus of an annulus if a biholomorphic map that uniformizes to a round annulus sends to a concentric round subannulus of ). Then all preimages of are annuli that map to by unbranched covering maps of degrees at most . One of them, say , is an essential subannulus in , and thus . ∎
Remark.
In fact, in the previous lemma one can prove the stronger bound , see [KL2, Lemma 4.5].
Lemma 2.5 (First entry maps have uniformly bounded degrees).
Let be a holomorphic map with a welldefined Markov partition. Then there exists a constant with the following property: for every puzzle piece of any depth and for every point that does not belong to the puzzle boundary of any depth, if is the least index so that , then the map has degree bounded by (independently of and ).
Proof.
Consider the sequence of puzzle pieces . We claim that these puzzle pieces have disjoint interiors. If not, then by the Markov property (Definition 2.1) we have for some and , hence , so in contradiction to minimality of (see Figure 2).
In particular, each critical point of can lie in the interior of at most one puzzle piece in this sequence. Therefore, the claim follows with equal to the product of the degrees of all critical points of . ∎
The following lemma, in a similar manner as Lemma 2.5, gives us control over the degree of the first entry map to the union of puzzle pieces that contains all critical puzzle pieces. We will say that a fiber is critical if it contains a critical point.
Lemma 2.6 (First entry to union of critical puzzle pieces has uniformly bounded degree).
Let be a finite set of points with distinct fibers which includes all critical fibers of . Suppose that there exists a depth so that all puzzle pieces of depth are pairwise disjoint, and an integer so that all are nondegenerate annuli. Then there is a constant with the following property: for every for which there exists a so that , let be minimal with this property; then there exists an essential open annulus such that .
Proof.
Consider an arbitrary for which there exists a so that , and suppose again that is minimal with this property, and that is not on the boundary of a puzzle at any depth. To fix notation, suppose that is a point in with .
We claim that then the set contains an annulus that separates from and that has modulus bounded below.
We have and hence , and by hypothesis this is a nondegenerate annulus of some modulus, say .
Now we take a preimage of this annulus under . The map sends to the puzzle piece at depth , and this is a branched cover of degree bounded in terms of and and the degrees of the critical points of .
Therefore,
will in general not be an annulus, but an open disk with several closed disks removed. However, it does contain an annulus that separates from , and that is an essential annulus with modulus bounded below in terms of , , and the degrees of the critical points of . Since there are only finitely many , this modulus is bounded below by a number that depends on , , and the set , but not on .
Now consider a point for which there exists a minimal as required in the lemma, and so that is not on the puzzle boundary at any depth. If , then is one of the discussed earlier.
The proof for the case is similar to Lemma 2.5. Again, consider the “orbit of puzzle pieces” . For , the point does not visit any critical puzzle piece of depth . Since for , the depth of the surrounding pieces exceeds , the entire pieces are noncritical. Therefore, the map
is biholomorphic. In particular, is conformally equivalent to
The claim now follows from the first part, applied to and . ∎
The limit set of a point is the set
Note that .
Lemma 2.7 (Accumulation on periodic fiber implies trivial fiber).
Let be a holomorphic map with a welldefined Markov partition. Suppose that is a nonescaping point of so that the limit set of intersects the fiber of some periodic point but the orbit of is disjoint from . Assume additionally that , as well as all those critical fibers of that intersect , are compactly contained in the corresponding puzzle pieces of any depth. Then .
Proof.
Our proof goes along the lines of the proof of [RY, Lemma 3], except for the final step where Lemma 2.6 will provide us with the suitable annuli to run the pullback argument.
The proof itself may look a bit technical in notation, but the underlying idea is simple: as long as the orbit of stays sufficiently close to , that is in some fixed puzzle piece that contains no further critical points other than those are already in , the puzzle pieces along this orbit are mapped forward injectively. When the orbit leaves and later returns back ( accumulates on by hypothesis), it does so with uniformly bounded degree by Lemma 2.5. This allows us, by pulling back suitable annuli (given by Lemma 2.6), to conclude that , whether or not the fiber of is trivial.
Up to passing to an iterate of , assume that is a fixed point, and let us adopt the notation for this iterate of ; thus for all and .
Let be the set of all critical fibers of , different from , that intersect (if the fiber of is not critical, then this is just the set of all critical fibers of that intersect ; here is some finite index set, which in the simplest case might be empty). For every critical fiber pick a critical point representing this fiber (this choice might not be unique). Let us choose so that and for every ; this is possible by definition of a fiber. By increasing if necessary, we also assume that does not intersect any critical puzzle piece of depth except those around , and, possibly, . Up to an index shift, assume . Further on, fix a depth such that all the annuli and are nondegenerate. The depth exists by the assumption of the lemma: all the fibers and are compactly contained in the corresponding puzzle pieces of any depth. Define ; this is the union of the puzzle pieces of depth containing and all critical fibers on which the orbit of accumulates.
For given , let be the smallest integer such that ; such an index exists because accumulates on . However, since the orbit of never enters by hypothesis, there exists a smallest integer such that , hence . Finally, let be minimal so that ; again, such an index exists because accumulates on , critical fibers , , and by the choice of what we call the zero depth puzzle pieces. However, it might happen that lands not in but in a critical puzzle piece in ; denote by the point from the set such that (see Figure 3 for a schematic drawing of the puzzle pieces involved).
We claim that there exists an essential open subannulus
such that
(2.1) 
where is given by Lemma 2.5 and is given by Lemma 2.6, and hence the factor is independent of and . We will do this in three steps; since we are pulling back, they come in reverse order. The third step is controlled by Lemma 2.5; the second step is a sequence of conformal iterates to ; and in the first step this puzzle piece is sent by to , controlled by Lemma 2.6 again. These three steps are illustrated in Figure 3 (left, center, and right); the annuli we are pulling back are contained in the shaded rings.
Step 1. Since was chosen to be minimal so that , and is such that , Lemma 2.6 guarantees that there exists an essential open subannulus
such that
(2.2) 
where does not depend on and . (Strictly speaking, in order to apply Lemma 2.6, we have to enlarge so that it would contain all critical puzzle pieces of depth ; but since, by construction, the orbit of visits only those critical puzzles already in , this enlargement of does not alter the conclusion.)
Step 2. We claim that there exists an open essential subannulus
such that is a conformal copy of , and hence
(2.3) 
We argue as follows. The puzzle pieces around are, as always, nested like , and since is a fixed point, each one is the image of the next one under . Since , all the points , , …, are in . But by construction all critical points in are already in and hence in . Therefore, for , the puzzle pieces of depth around do not contain critical points. Together, this shows that the map has degree , and the same is true for its restriction , and hence the claim in Step 2 follows with as the conformal pullback of under this restricted map.
Step 3. Similarly to Step 1, since is the first iterate so that , the map has degree at most by Lemma 2.5. The same is then true for its restriction
Let be a preimage of the annulus under this restricted map chosen in such a way that is an essential subannulus in . By Lemma 2.4,
(2.4) 
This argument can be carried out for infinitely many : we choose a sequence so that once is fixed, the value of is chosen so that is contained in the bounded component of . This way, we obtain infinitely many disjoint annuli with moduli bounded below that all separate from all previous annuli, and using the standard Grötzsch inequality this implies that the fiber of is trivial. ∎
We say that a puzzle piece of depth is weakly protected if there exists a puzzle piece of depth such that is compactly contained in . If , then is protected. The following lemma guarantees compact containment of pullbacks of certain weakly protected puzzle pieces.
Lemma 2.8 (First return to weakly protected puzzle piece, recurrent case).
Let be a holomorphic map with a welldefined Markov partition. Suppose that is a point that does not belong to the puzzle boundary of any depth and for which there exists a with , and suppose further that the minimal such has the property that ; in particular, is weakly protected. Then every component of the set is compactly contained in .
Proof.
By the Markov property, the closure of any component of the set is a puzzle piece of depth at least . Let be one of these puzzle pieces. If with is the depth of , then is a branched covering. Since is the first return time of the orbit of back to , the puzzle pieces have disjoint interiors. Therefore, .
Suppose that is not compactly contained in . Then is not compactly contained in , and there exists a point . But since puzzle pieces are nice sets, i.e. for all , we have in particular . But we must have , a contradiction. ∎
Lemma 2.9 (First return to weakly protected puzzle piece, nonrecurrent case).
Let be a holomorphic map with a welldefined Markov partition. Suppose that is a combinatorially nonrecurrent point that does not belong the puzzle boundary of any depth. Let be a depth such that the orbit of never returns to . If there exists such that is weakly protected by , then every component of the set is compactly contained in .
Proof.
Similarly to the proof of Lemma 2.8, the closure of every component in the set is a puzzle piece of depth at least ; let be such a puzzle piece, and with be its depth. Since the orbit of never returns to , it follows that for every the puzzle piece is disjoint from . The same is true for because ; hence . Finally, since is weakly protected by , and , by pulling back we conclude that the puzzle piece is weakly protected by the puzzle piece of depth . Since , this puzzle piece lies in . Therefore, is compactly contained in . ∎
Remark.
It is possible to show that if is protected, then every component of the first return domain to is compactly contained in , see [L2, ].
3. Complex box mappings and rigidity (Theorem C)
In this section we review the notion of complex box mappings introduced in [KSS, KvS1] (with some further clarification in [KvS2]) and prove a generalized version of trivial fibers for such mappings (Theorem C). This result is of interest in its own right, and it is a key ingredient in the proof of our Rational Rigidity Principle (Theorem A).
Definition 3.1 (Complex box mapping [KvS1, KvS2]).
A holomorphic map between two open sets is a complex box mapping if the following holds:

has finitely many critical points;

is the union of finitely many open Jordan disks with disjoint closures;

every component of is either a component of , or is a union of Jordan disks with pairwise disjoint closures, each of which is compactly contained in ;

for every component of the image is a component of , and the restriction is a proper map.
Following [DH2], a proper holomorphic map of degree between two open topological disks and with is called a polynomiallike map (in the sense of Douady–Hubbard [DH2]). By the straightening theorem, such a polynomiallike map is hybrid equivalent to a polynomial of degree , and this polynomial is unique (up to affine conjugation) if the filled Julia set is connected. Moreover, connectivity of is equivalent to the condition that all critical points of are contained in .
When is connected and has only finitely many components, and all of these are compactly contained in , then the corresponding box mapping may be regarded as a polynomiallike map in the sense of Douady–Hubbard (generalized to several components of ). For general box mappings, however, is allowed to have infinitely many components, and in many applications this is important. Such generality in the definition of a box mapping results in phenomena that do not occur for polynomiallike maps. For example, a box mapping might wandering domains, or it might have a filled Julia set that is all of ; see below, as well as [KvS2].
Definition 3.2 (Puzzle piece of box mapping).
For a box mapping , we define a puzzle piece of depth to be a component of . A puzzle piece is called critical if it contains at least one critical point.
The set is the filled Julia set of the box mapping ; this is the set of nonescaping points. Similarly, the Julia set is defined as .
Definition 3.3 (Box renormalizable box mappings).
We call a complex box mapping box renormalizable around a critical point if there exists a puzzle piece at some depth containing , and an integer (called the period of the renormalization) such that for every critical point and every , and is minimal with this property. The filled Julia set of this box renormalization is defined analogously as . In this context we call a box renormalizable critical point.
A complex box mapping is called box renormalizable if it is box renormalizable around at least one critical point in , and nonbox renormalizable otherwise.
Remark.
If we denote by the component of containing , then either , or .
If is compactly contained in (that is ), then the restriction is a polynomiallike map in the sense of Douady–Hubbard, and we simply say that is renormalizable. Moreover, the filled Julia set of the renormalization around is connected, by the standard theory of polynomiallike maps mentioned above.
In the case that a puzzle piece contains several critical points among which some have their entire orbits in and others do not, then one can shrink to a puzzle piece of greater depth that contains only those critical points that do not escape, and then is renormalizable around these critical points.
If , then is a proper selfmap of a disk without escaping points, and hence the filled Julia set of , restricted to , is equal to . This is a “pathological case” that may occur for box renormalizable maps, but it is not included in our definition of renormalization; it does not occur in a number of interesting cases arising from dynamics on . Our definition of box renormalizability of a box map coincides with the notion of renormalizability as used in [KvS1, Definition 1.3]).
Remark.
If is a renormalizable critical point, and is a corresponding to polynomiallike map, then the filled Julia set of is, in fact, equal to (the set of all points with the same periodic itinerary). To see this, first observe that the fiber of any nonescaping point in under is contained in the filled Julia set of . Moreover, each of such fibers is compactly contained in every puzzle piece of any depth. For a given nonescaping point , if , then the claim follows. Otherwise, there exists a pair of separating puzzle pieces and of the same depth with and constructed as pullbacks of . Since the filled Julia set of is connected and contains both and , it should then intersect the boundaries of and . But the points on the boundaries of and escape under , a contradiction.
We will be using the following theorem by Kozlovski and van Strien [KvS1, Theorem 1.4] (with some clarification in [KvS2]); this theorem plays a crucial role in their study of rigidity for multicritical complex and real polynomials (see [KSS, KvS1]; see also [KL2] for the original proof of the Kahn–Lyubich Covering Lemma, a crucial technical ingredient used to obtain the allimportant complex bounds).
Theorem 3.4 (Rigidity for complex box mappings [KvS1, KvS2]).
Assume that is a nonrenormalizable complex box mapping for which all periodic points are repelling. Then each point in its Julia set has trivial fiber or converges to the boundary of . ∎
Lemma 3.5 (Fibers compactly contained in puzzle pieces).
Consider a box mapping and a nonescaping point . If the orbit of does not eventually land in a cycle of periodic components without escaping points (LABEL:Item:NE), then the fiber of is compactly contained in any puzzle piece it is contained in.
Proof.
It suffices to prove that every puzzle piece around compactly contains another puzzle piece around at greater depth. To do this, let be the puzzle piece of any depth containing and for denote by the puzzle piece around at depth , so that is a component of .
If any contains a puzzle piece around at greater depth than that is a proper subset, then this proper subset must be compactly contained, and the claim follows by pullback to . Otherwise, in particular is not only a component of but also a component of . As we iterate forward, we cannot keep visiting components of that are also components of (by finiteness of this would yield a cycle of (LABEL:Item:NE)components which is excluded by hypothesis), so we must reach a component of that is compactly contained in its component of , and the claim follows. ∎
Proof of the generalized rigidity principle (Theorem C).
Let be a complex box mapping, and let be a nonescaping point. The claim of the theorem is that then at least one of the following holds: has (LABEL:Item:T) trivial fiber, or it has (LABEL:Item:R) renormalizable dynamics, or it has “pathological dynamics” in the sense that (LABEL:Item:NE) it eventually maps to a periodic component without escaping points, or (LABEL:Item:CB) the orbit converges to .
It thus suffices to consider a nonescaping point to which neither (LABEL:Item:NE) nor (LABEL:Item:CB) apply. We need to show that satisfies either (LABEL:Item:T) or (LABEL:Item:R). By Lemma 3.5, the fiber is compactly contained in puzzle pieces of any depth, and absence of (LABEL:Item:CB) implies that .
Denote by be the set of all periodic fibers of that are renormalizable in the sense that they are contained in a puzzle piece of level and so that is a polynomiallike map with connected Julia set and of degree at least . These puzzle pieces can be chosen to be disjoint for different renormalizable fibers, and each of them contains at least one critical point of . Therefore, contains only finitely many fibers. All their Julia sets contain at least one fixed point of .
For the given point , we have the following three possibilities:

there exists an such that ;

for all , but there exists an index such that ;

does not intersect the fibers in .
Let us make several remarks concerning this case distinction. Since the components of have disjoint closures and are compactly contained in the respective components of (unless they coincide), all fibers of are disjoint. Therefore, in the first case above the index defines a unique cycle of fibers. However, in the second case the index might be not unique; for us it is enough to have at least one such index. Note that it is impossible that the orbit of intersects one fiber in and accumulates on another fiber since the fibers in are periodic and disjoint.
Case (1) is exactly possibility (LABEL:Item:R) in the statement of Theorem C, so in this case we are done. It remains to show that in the other two cases, the fiber of is trivial.
Let us now treat case (2). The fiber contains a periodic point, but the orbit of never lands in a periodic fiber. This is the situation of Lemma 2.7; we explain why the assumptions in this lemma are satisfied. The fiber is compactly contained in all its puzzle pieces by the condition on renormalizability. Moreover, since the orbit of never lands in an (LABEL:Item:NE) component by the hypothesis, each critical fiber on which it can further accumulate is compactly contained in any surrounding puzzle piece of any depth by Lemma 3.5 (note that, by the definition of a fiber, all such fibers must contain a nonescaping critical point; moreover, the orbit of this nonescaping critical point never lands in an (LABEL:Item:NE) component). Therefore, the assumptions of Lemma 2.7 are satisfied, and thus the fiber of is trivial.
In order to tackle case (3), we will show that the orbit of is eventually contained in the Julia set of a nonrenormalizable box mapping with only repelling periodic points. The map will be obtained as a suitable restriction of (iterates of) , and the conclusion will then follow from Theorem 3.4.
Let be the set of all critical points the orbits of which either land in, or accumulate at, one of the periodic fibers in . Also define to be the set of all critical points of that eventually map to components of of type (LABEL:Item:NE) (periodic components without escaping points); by construction, . Periodic critical points in are box renormalizable but not renormalizable in our (i.e. the Douady–Hubbard) sense.
Finally, write . The critical points in are not box renormalizable. Consider the union of the interiors of critical puzzle pieces of the same sufficiently large depth such that:

each contains at least one point in