Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
aa r X i v : . [ m a t h . D S ] M a y BIFURCATION LOCI OF EXPONENTIAL MAPSAND QUADRATIC POLYNOMIALS:LOCAL CONNECTIVITY, TRIVIALITY OF FIBERS,AND DENSITY OF HYPERBOLICITY
LASSE REMPE AND DIERK SCHLEICHER
Abstract.
We study the bifurcation loci of quadratic (and unicritical) polynomialsand exponential maps. We outline a proof that the exponential bifurcation locus isconnected; this is an analog to Douady and Hubbard’s celebrated theorem that (theboundary of) the Mandelbrot set is connected.For these parameter spaces, a fundamental conjecture is that hyperbolic dynam-ics is dense. For quadratic polynomials, this would follow from the famous strongerconjecture that the bifurcation locus (or equivalently the Mandelbrot set) is locallyconnected. It turns out that a formally slightly weaker statement is sufficient, namelythat every point in the bifurcation locus is the landing point of a parameter ray.For exponential maps, the bifurcation locus is not locally connected. We describe adifferent conjecture (triviality of fibers) which naturally generalizes the role that localconnectivity has for quadratic or unicritical polynomials. Bifurcation Loci and Stable Components
The family of quadratic polynomials p c : z z + c , parametrized by c ∈ C , contains,up to conformal conjugacy, exactly those polynomials with only a single, simple, criticalvalue (at c ). Hence this family is the simplest parameter space in the dynamical studyof polynomials, and has correspondingly received much attention during the last twodecades. Similarly, exponential maps E c : z e z + c are, up to conformal conjugacy,the only transcendental entire functions with only one singular value (the asymptoticvalue at c ). This simplest transcendental parameter space has likewise been studiedintensively since the 1980s.In the following, we will treat these parameter spaces in parallel, unless explicitlystated otherwise; we will write f c for p c or E c . Following Milnor, we write f ◦ nc for the Mathematics Subject Classification.
Key words and phrases.
Quadratic polynomial, exponential map, Mandelbrot set, bifurcation locus,parameter space, parameter ray, local connectivity.We gratefully acknowledge that this work was supported by the European Marie-Curie researchtraining network CODY, the ESF research networking programme HCAA, and EPSRC fellowshipEP/E052851/1. Often the parametrization z λe z with λ ∈ C ∗ is used instead. This has the asymptotic valueat 0 and is conformally equivalent to E c iff λ = e c . This has the advantage that two maps z λe z and z λ ′ e z are conformally conjugate iff λ ′ = λ . We prefer the parametrization as E c not onlyfor the analogy to the quadratic family, but also because all maps E c have the same asymptotics nearinfinity, and because parameter space is simply connected, which leads to a more natural combinatorialdescription. (a) p c (b) E c Figure 1.
The parameter spaces of quadratic polynomials p c (left) andexponential maps E c (right); the bifurcation loci are drawn in black. Un-bounded hyperbolic components are white, bounded hyperbolic compo-nents are gray (these do not exist for exponentials). n -th iterate of f c . The map f c is called stable if, for c ′ sufficiently close to c , the maps f c and f c ′ are topologically conjugate on their Julia sets, and the conjugacy dependscontinuously on the parameter c ′ (the former condition implies the latter in our setting).We denote by R the locus of stability ; that is, the (open) set of all c ∈ C so that f c isstable. The set R is open and dense in C [MSS, EL].A hyperbolic component is a connected component of R in which every map f c hasan attracting periodic orbit of constant period. Within any non-hyperbolic componentof R , all cycles of f c would have to be repelling. One of the fundamental conjectures ofone-dimensional holomorphic dynamics is the following. Conjecture 1 (Hyperbolicity is Dense) . Every component of R is hyperbolic. (Equiva-lently, hyperbolic dynamics is open and dense in parameter space.) Hyperbolic components — both in the quadratic and in the exponential family — arecompletely understood in terms of their combinatorics [DH, S2, RS1]. The complement B := C \ R is called the bifurcation locus . Since R is open and dense in C , B is closedand has no interior points. The bifurcation locus is extremely complicated (see Figure1). Theorem 2 (Bifurcation Loci Connected) . B is a connected subset of C . For quadratic polynomials, B is the boundary of the famous Mandelbrot set M , andTheorem 2 is equivalent to the fundamental theorem of Douady and Hubbard that M orequivalently ∂ M is connected [DH, Expos´e VIII.I]. For exponential maps, connectivityof the bifurcation locus is new [RS1, Theorem 1.1]; we outline a proof below. IFURCATION LOCI 3
To study the bifurcation locus, it is useful to consider the escape locus I := { c ∈ C : f ◦ nc ( c ) → ∞ as n → ∞} . The set I decomposes naturally into a disjoint union of parameter rays and their end-points (see below), and the ultimate goal is to describe B in terms of these rays. Thestructure of I and the parameter rays is well-understood, and conjecturally, every pointin B is the landing point of a parameter ray or, in the case of exponential maps, ona parameter ray itself. This is in analogy to the dynamical planes of f c , where theJulia sets are often studied using the simpler structure of the Fatou set or of the setof points that converge to ∞ under iteration. We show below (Theorem 11) that, forthe quadratic family, the conjecture that every point in B is the landing point of a rayis equivalent to the famous open question of local connectivity of the Mandelbrot set;we then reformulate this conjecture in a uniform way for quadratic polynomials andexponentials.A fundamental difference between the polynomials p c and the exponential maps E c istheir behavior near ∞ : every p c has a superattracting fixed point at ∞ which attractsa neighborhood of ∞ in the Riemann sphere, while every E c has an essential singularityat ∞ and the set of points converging to ∞ is extremely complicated [SZ]. This impliesthat in the parameter space of quadratic polynomials we have I = C \ M and there isa unique conformal isomorphism Φ : I → C \ D with Φ( c ) /c → c → ∞ (where D is the complex unit disk); the map Φ was constructed by Douady and Hubbard [DH,Expos´e VIII.I] in their proof of connectivity of the Mandelbrot set. A parameter ray isdefined as the preimage of a radial line in C \ D under Φ. On the other hand, the set I for exponential maps has a much richer topological structure [FRS]: it is the disjointunion of uncountably many curves ( parameter rays ) with or without endpoints; eachparameter ray (possibly with its endpoint) is a path component of I . More precisely,every path component of I is a curve G s : (0 , ∞ ) → I or G s : [0 , ∞ ) → I , both timeswith G s ( t ) → ∞ as t → ∞ . The index s distinguishes different parameter rays: itis a sequence of integers and is called the external address of G s . Different rays havedifferent external addresses, and the set of allowed external addresses can be describedexplicitly [FS, FRS]. We call the image G s (0 , ∞ ) the parameter ray at external address s , and G s (0) its endpoint (if it exists). Let I R be the union of all parameter rays and I E be the set of all endpoints in I . We say that a parameter ray lands if lim t ց G s ( t )exists (note that many parameter rays G s land at non-escaping parameters, and othersmight not land at all, but I E consists only of those landing points that are in I ).It is a consequence of the “ λ -lemma” from [MSS] that B = ∂ I both for quadraticpolynomials and for exponential maps; see e.g. [R1, Lemma 5.1.5]. In particular, thebifurcation locus of quadratic polynomials is a compact subset of C , while for exponentialmaps it is unbounded.We will also use the reduced bifurcation locus B ∗ := B \ I R ;for quadratic polynomials, clearly B ∗ = B , but for exponential maps, B ∗ is a propersubset of B . The parametrization in [FRS] is somewhat different, but this is of no consequence in the following.
LASSE REMPE AND DIERK SCHLEICHER
We should remark also on the family of unicritical polynomials: those which have aunique critical point in C . Such polynomials may be viewed as the topologically andcombinatorially simplest polynomials of a given degree d , and they are affinely conjugateto p d,c : z z d + c or to z λ (1 + z/d ) d with λ = dc d − . These unicritical polynomi-als are often viewed as combinatorial interpolation between quadratic polynomials andexponential maps; see [DGH, S5, S4]. Everything we say about quadratic polynomi-als remains true also for unicritical polynomials, but for simplicity of exposition andnotation we usually speak only of quadratic polynomials and of exponentials. Structure of the article.
In Section 2, we review the famous “MLC” conjecture forthe Mandelbrot set, and then define “fibers” (introduced for the case of M in [S4])of quadratic polynomials and exponential maps in parallel. This concept allows us toformulate Conjecture 8 on triviality of fibers, which is equivalent to MLC in the settingof quadratic polynomials. We also discuss a number of basic results on fibers. For easeof exposition, the theorems stated in Section 2 will be proved, separately, in Section 3.Apart from a few somewhat subtle topological considerations, the proofs are not toodifficult, but rely on a number of recent non-trivial results on the structure of exponentialparameter space. We cannot comprehensively review all of these in the present article,but have attempted to present the proofs so that they can be followed without detailedknowledge of these papers.While most of the results stated are well-known in the case of the Mandelbrot set,some observations seem to be new even in this case. (Compare, in particular, Theorem11, which allows a simple, and formally weaker, restatement of the MLC conjecture.)2. Local Connectivity and Trivial Fibers
Local connectivity of bifurcation loci.
It was conjectured by Douady and Hubbardthat the Mandelbrot set M is locally connected; this is perhaps the central open problemin holomorphic dynamics. The following is an equivalent formulation. Conjecture 3 (MLC) . The quadratic bifurcation locus B = ∂ M is locally connected. One of the reasons that this conjecture is important is that it implies Conjecture 1for quadratic polynomials: see Douady and Hubbard [DH, Expos´e XXII.4] (see also [S4,Corollary 4.6], as well as Theorems 9 and 10 below).
Theorem 4 (MLC Implies Density of Hyperbolicity) . If the Mandelbrot set is locallyconnected, then hyperbolic dynamics is dense in the space of quadratic polynomials.
In topological terms, the situation in exponential parameter space is very differentfrom what we expect in the Mandelbrot set: the analog of Conjecture 3 is known to befalse.
Theorem 5 (Failing Local Connectivity of Exponential Bifurcation Locus) . The ex-ponential bifurcation locus B is not locally connected. More precisely, B is not locallyconnected at any point of I R . In essence, failure of local connectivity of B in the exponential setting was first shownby Devaney: from his proof [De] that the exponential map exp itself is not structurallystable, it also follows that B is not locally connected at the point c = 0. Theorem IFURCATION LOCI 5
Figure 2.
Exponential parameter space contains “Cantor bouquets”,which are closed sets consisting of uncountably many disjoint simplecurves. Some such curves are indicated here in black.5 is related to the existence of so-called
Cantor bouquets within the bifurcation locus;compare Figure 2. These consist of uncountably many curves (on parameter rays) thatare locally modelled as a subset of the product of an interval and a Cantor set.
Fibers.
Local connectivity of the Mandelbrot set would have a number of importantconsequences apart from that described by Theorem 4: for example, there are a numberof topological models for M (such as Douady’s pinched disks [Do] or Thurston’s qua-dratic minor lamination [T]) that are homeomorphic to M if and only if M is locallyconnected. Moreover, MLC is equivalent to the combinatorial rigidity conjecture : anytwo maps in B without indifferent periodic orbits can be distinguished combinatorially(in terms of which periodic dynamic rays land together). Hence it is desirable to findanother topological concept which can play the role of local connectivity in the spaceof exponential maps. One convenient notion of this type is provided by triviality offibers , introduced for the Mandelbrot set in [S4]; it has the advantage of transferringeasily to exponential parameter space. Another advantage is that even for polynomials,any possible failure of local connectivity can be made more precise by giving topologicaldescriptions of non-trivial fibers. Definition 6 (Separation Line and Fiber) . A separation line is a Jordan arc γ ⊂ C in parameter space, tending to ∞ in both directions and containing only hyperbolic andfinitely many parabolic parameters . Since all hyperbolic components of exponential parameter space are unbounded and ∞ is accessible,it suffices to allow just a single parabolic parameter on every separation line; for M at most two parabolicparameters suffice. LASSE REMPE AND DIERK SCHLEICHER
Figure 3.
A separation line in the space of exponential maps
A separation line γ separates two points c, c ′ ∈ B ∗ if c and c ′ are in two differentcomponents of C \ γ .The extended fiber of a point c ∈ C is the set of all c ′ which cannot be separated from c by any separation line. The (reduced) fiber of c ∈ C \ I R consists of all c ′ in theextended fiber of c which do not belong to I R .A fiber is called trivial if it consists of exactly one point.Remark 1. This definition is somewhat different from (but equivalent to) that originallygiven in [S4] for quadratic polynomials. See the remark on alternative definitions ofseparation lines below.
Remark 2.
One fundamental difference between polynomial and exponential parameterspace is that the escape locus I is hyperbolic only in the polynomial case; for exponentialparameter space, separation lines are disjoint from I and thus from parameter rays (analternative definition of separation lines uses parameter rays; see below). Remark 3. If c is a hyperbolic parameter, then it follows easily from the definition thatthe fiber of c is trivial. Hence all interest lies in studying non-hyperbolic fibers, and wewill usually restrict our attention to this case. Theorem 7 (Properties of fibers) . Every extended fiber ˘ Y is a closed and connectedsubset of parameter space. Either ˘ Y is a single point in a hyperbolic component, or ˘ Y ∩ B ∗ = ∅ .In particular, ˘ Y contains the (reduced) fiber Y = ˘ Y \ I R ; this fiber is either trivial oruncountable.Remark 1. For fibers in the Mandelbrot set, these claims are immediate from the defi-nition. For exponential maps, the proofs are more subtle, and require some topologicalconsiderations as well as rather detailed knowledge of parameter space.
IFURCATION LOCI 7
Remark 2.
Once we know that parabolic parameters have trivial fibers (which is well-known for quadratic polynomials [S4, TL, H]; for exponential maps, a proof will beprovided in [R4]), it also follows that fibers are pairwise disjoint.Armed with the definition of fibers, we can now propose the following central conjec-ture.
Conjecture 8 (Fibers are trivial) . For the spaces of quadratic polynomials and expo-nential maps, all fibers are trivial.
Since any non-hyperbolic stable component would be contained in a single fiber, itfollows immediately that triviality of fibers would settle density of hyperbolicity:
Theorem 9 (Triviality of fibers implies density of hyperbolicity) . Conjecture 8 impliesConjecture 1. (cid:4)
Triviality of fibers and local connectivity of the Mandelbrot set.
FollowingMilnor [M, Remark after Lemma 17.13], we say that a topological space X is locallyconnected at a point x ∈ X if x has a neighborhood base consisting of connected sets. Triviality of the fiber of a parameter c in the quadratic bifurcation locus implies localconnectivity of the Mandelbrot set at c . In fact, all known proofs of local connectivityof M at given parameters do so by actually establishing triviality of fibers.The converse question — in how far local connectivity implies triviality of fibers —is more subtle. For example, a full, compact, connected set K may well contain non-accessible points at which K is locally connected; see Figure 4(a). On the other hand,triviality of the fiber of a parameter c implies that there is a parameter ray landing at c (see Theorem 11 below), and hence that c is accessible from the complement of M .So local connectivity at c does not formally imply triviality of the fiber of c , but thereis the following, more subtle, connection. Theorem 10 (Trivial Fibers and MLC) . Let c belong to the quadratic bifurcation locus B . Then the following are equivalent: (a) the Mandelbrot set M is locally connected at every point of the fiber of c ; (b) the fiber of c is trivial.In particular, local connectivity of the Mandelbrot set is equivalent to triviality of allfibers in the quadratic bifurcation locus. Triviality of Fibers and Landing of Parameter Rays.
For the Mandelbrot set,by Carath´eodory’s theorem [M, Theorem 17.14] local connectivity implies that everyparameter ray lands, and the map assigning to every external angle the landing point ofthe corresponding parameter ray is a continuous surjection S → B .Again, replacing local connectivity by triviality of fibers allows us to obtain a state-ment regarding the landing of rays which is true in both quadratic and exponentialparameter space. Theorem 11 (Landing of rays implies triviality of fibers) . Let c ∈ B ∗ . Then the follow-ing are equivalent. Sometimes this property is instead referred to as “connected im kleinen” (cik) at x , and the term“locally connected at x ” is reserved for what Milnor calls “openly locally connected at x ”. LASSE REMPE AND DIERK SCHLEICHER (a) (b)
Figure 4.
Examples of compact connected sets K ⊂ C with connectedcomplement such that (a) K is locally connected at a point z ∈ K , but z is not accessible from C \ K (where z is the point at the center of thefigure); (b) every z ∈ K is accessible from C \ K (and hence the landingpoint of an external ray), but K is not locally connected.(a) The fiber of c is trivial. (b) Every point in the fiber of c is the landing point of a parameter ray.In particular, triviality of all fibers is equivalent to the fact that every point of B ∗ isthe landing point of a parameter ray. The last sentence in the theorem provides a convenient way of stating Conjecture 8without the definition of fibers.The previous result leads to an interesting observation about the Mandelbrot set,which is new as far as we know:
MLC is equivalent to the claim that every c ∈ ∂ M isaccessible from C \ M . For general compact sets, this is far from true (compare Figure4(b)).In this context, there is a difference between the situation for the Mandelbrot set andthat of exponential parameter space. In the former case, triviality of the fiber of c impliesthat all parameter rays accumulating on c land. In the latter case, we can only concludethat one of these rays lands. The problem is that escaping parameters are contained inthe bifurcation locus; it is compatible with our current knowledge that one parameterray might accumulate on another ray together with its landing point (compare Figure5), in which case the corresponding fiber could still be trivial. This might lead us toformulate a stronger variant of Conjecture 8: all fibers are trivial, and furthermore allparameter rays land . It seems plausible that these conjectures are equivalent (we believethat both are true). Alternative definitions of separation lines.
We defined separation lines as curvesthrough hyperbolic components. We did so since this gives a simple definition and doesnot require the landing of periodic parameter rays in the exponential family (a fact thatwas proved in [S1], which is however not formally published).There are a number of other definitions we could have chosen; for example,
IFURCATION LOCI 9 c G G Figure 5.
It is conceivable that an extended fiber in exponential param-eter space consists of one ray G landing at some parameter c togetherwith a second ray G which accumulates not only on c , but also on asegment of G . In this case, the fiber of c would be trivial, but the ray G clearly does not land at c .(a) A separation line is a curve consisting either of two parameter rays landing ata common parabolic parameter, or of two parameter rays landing at distinctparabolic parameters, together with a curve which connects these two landingpoints within a single hyperbolic component.(b) A separation line is a curve as in (a), except that we also allow two parameterrays landing at a common parameter for which the singular value is preperiodic.(c) A separation line is a Jordan arc, tending to infinity in both directions, containingonly finitely many parameters which are not escaping or hyperbolic.These alternative definitions will yield a theory of fibers for which all of the aboveresults remain true. However, it is a priori conceivable that separation lines run throughparameters with nontrivial fibers, in which case the fiber of a nearby parameter c ∈ B ∗ may depend on the definition of separation lines . On the other hand, the question of triviality of such a fiber, and hence Conjecture 8, is independent of this definition.3. Proofs
We begin by showing that the exponential bifurcation locus is not locally connected.
Proof of Theorem 5. (Exponential bifurcation locus not locally connected.)
As alreadyremarked, failure of local connectivity in the exponential setting follows already fromDevaney’s proof of structural instability of exp [De]. Essentially, he showed that theinterval [0 , ∞ ) ⊂ B is accumulated on by curves in hyperbolic components (compareFigure 6).We will now indicate how to prove failure of local connectivity at every point of I R using a similar idea, together with more detailed knowledge of exponential parameter The original definition in [S4] for M is (b), which is shown to be equivalent to (a); for M , allparabolic parameters have trivial fibers. Figure 6.
Curves in hyperbolic components accumulating on the interval[ − , ∞ ) in exponential parameter space.space. More precisely, let G s : (cid:0) , ∞ (cid:1) → C be a parameter ray in exponential parameterspace. We claim that, for every t >
0, the curve G s : [ t , ∞ ) → C is the uniform limitof curves γ n : [ t , ∞ ) → C within hyperbolic components W n of period n .Every small neighborhood U of a parameter G s ( t ) with t > t then intersects theboundaries of infinitely many of these components, and these boundaries are separatedfrom G s ( t ) within U by the curves γ n . This proves the theorem.In the case where t is sufficiently large, the existence of the curves γ n is worked outin detail in [RS1, Section 4]. Here, we will content ourselves with indicating the overallstructure of the proof, for arbitrary t > c = G s ( t ) for t ≥ t , and let z n := E n ( c ) := E ◦ nc ( c ) denote the singular orbitof E c . Since c is on a parameter ray, the real parts of the z n converge to infinity likeorbits under the (real) exponential function. More precisely, we have the asymptotics z n = F ◦ n ( T ) + 2 πis n +1 + o (1) , where F ( x ) = exp( x ) − s = s s s . . . and T is some positive real number. Given theexpansion of exp in the right half plane, it should not be surprising that (cid:12)(cid:12) ( E n ) ′ ( c ) (cid:12)(cid:12) → ∞ as n → ∞ (for a proof, see [BBS, Lemma 6]). Furthermore, this growth of the derivativeis uniform for t ≥ t .It follows readily (compare [BBS, Lemma 3]) that, for sufficiently large n , c = G s ( t )can be perturbed to a point c n = γ n ( t ) whose singular orbit follows that of E c closelyuntil the ( n − iπ . In other words, (cid:12)(cid:12) E k ( c n ) − z k (cid:12)(cid:12) ≪ k = 0 , , . . . , n − E n − ( c n ) = z n − + iπ. This implies that the real part of E ◦ n − c n ( c n ) is very negative, and hence E ◦ nc n ( c n ) isvery close to c . So the orbit of the singular value c n is “almost” periodic of period n . IFURCATION LOCI 11
The contraction along this orbit is such that a certain disk around c n is mapped intoitself, and E c n has an attracting periodic orbit of period n (similarly as in [BR, Lemma7.1]; compare [S2, Lemma 3.4]). It is easy to see that c n = γ n ( t ) is continuous in t andconverges uniformly to G s ( t ) for n → ∞ , as desired. (cid:4) Remark 1.
We remark that the curves γ n constructed in the proof converge to G s (cid:0) [ t , ∞ ) (cid:1) “from above”, in the sense that they tend to infinity in the unique component of { Re z > Re G s ( t ) } \ G s (cid:0) [ t , ∞ ) (cid:1) which contains points with arbitrarily large imaginary parts. In equation (*), we couldhave just as well chosen c n such that E n − ( c n ) = z n − − iπ ; in this case the curves γ n would converge to G s (cid:0) [ t , ∞ ) (cid:1) from below. We will use this fact in the proof of Lemma13 below. Remark 2.
We do not currently know anything about local connectivity of B at pointsof B ∗ . It seems conceivable that B is locally connected exactly at the points of B ∗ ; butin our view the more relevant question is whether all fibers of points in B ∗ are trivial.To begin our discussion of fibers, we note the following elementary consequences oftheir definition. Lemma 12 (Extended Fibers) . Every extended fiber ˘ Y is a closed subset of C ; theclosure ˆ Y of ˘ Y in the Riemann sphere is compact and connected.(In the space of quadratic polynomials, every extended fiber is bounded and hence ˆ Y =˘ Y ; in the space of exponential maps, every non-hyperbolic extended fiber is unboundedand hence has ˆ Y = ˘ Y ∪ {∞} .)Proof. Let ˘ Y be the extended fiber of a point c . The set of points not separated from c by a given separation line is a closed subset of C . So extended fibers are defined asan intersection of a collection of closed subsets of C , and therefore closed themselves.It is easy to verify that a point c which is separated from c by a collection of separationlines can also be separated from c by a single separation line. Furthermore, one can verifythat only countably many separation lines γ are necessary to separate c from all pointsoutside ˘ Y . For example, we can require that the intersection of γ with any hyperboliccomponent W is a hyperbolic geodesic of W . Since the set of parabolic parameters iscountable and every separation line runs through hyperbolic components and finitelymany parabolic parameters, there are only countably many such separation lines.It follows that ˘ Y can be written as a nested countable intersection of closed, connectedsubsets of the plane; this proves connectivity. (cid:4) Before we prove the remaining theorems, we require some preliminary combinatorialand topological considerations in exponential parameter space. These become necessarymainly because we need to take into account the possibility of parameter rays accumu-lating on points of I R .We begin by noting that there is a natural combinatorial compactification of exponen-tial parameter space, as follows. The external addresses considered so far are elements of Z N ; we will sometimes call these infinite external addresses. We also introduce interme-diate external addresses: these have the form s . . . s n − ∞ , where n ≥ s , . . . , s n − ∈ Z and s n − ∈ Z + 1 /
2; note that there is a unique intermediate external address of length n = 1, namely ∞ . The set of all infinite and intermediate external addresses will bedenoted S . The lexicographic order induces a complete total order on S \ {∞} and acomplete cyclic order on S ; compare [RS2, Section 2].We can then define a natural topology on ˜ C = C ∪ S , which has the property that G s ( t ) → s in this topology as t → + ∞ . The space ˜ C is homeomorphic to the closed unitdisk D , where C corresponds to the interior of the disk, and S to the unit circle. (Thisconstruction is analogous to compactifying parameter space of quadratic polynomials byadding a circle of external angles at infinity.) See also [RS2, Appendix A].For any parameter ray G s , we now set G s ( ∞ ) := s ∈ ˜ C , giving a parametrization G s : (0 , ∞ ] → ˜ C . Similarly, recall the curves γ n accumulating on the parameter ray G s as constructed in the proof of Theorem 5. They can be extended continuously by setting γ n ( ∞ ) = s n , where s n is an intermediate external address and s n → s . Armed with thisterminology, we can state and prove the following key fact, which we will use repeatedly. Lemma 13 (Accumulation on parameter rays) . Let A ⊂ ˜ C be connected, and let ˜ A be theclosure of A in ˜ C . Suppose that A intersects at most finitely many hyperbolic componentsand that A ∩ S contains at most finitely many intermediate external addresses.Suppose that G is a parameter ray, and that G ( t ) ∈ ˜ A for some t ∈ (0 , ∞ ] . Theneither ˜ A ⊂ G (cid:0) [ t, ∞ ] (cid:1) for some < t ≤ t , or G (cid:0) (0 , t ] (cid:1) ⊂ ˜ A .Proof. If G ((0 , t ]) ⊂ ˜ A , then nothing is left to prove, so we may assume that there isa t ∈ (0 , t ) with G ( t ) / ∈ ˜ A and then show that ˜ A ⊂ G ([ t, ∞ ]). There is some ε > D ε ( G ( t )) ∩ A = ∅ . Recall from the proof of Theorem 5 and the subsequent remarkthat G ([ t, ∞ )) is accumulated on from above resp. below by curves γ + n : [ t, ∞ ) → C and γ − n : [ t, ∞ ) → C , each contained in a hyperbolic component of period n . Also recall thatwe can extend these curves continuously by setting γ ± n := s ± n with suitable intermediateexternal addresses s ± n . We then have s + n ց s and s − n ր s .By assumption, A intersects at most finitely many of the curves γ ± n . So for sufficientlylarge n , A is disjoint from K n := D ε ( G ( t )) ∪ γ + n (cid:0) [ t, ∞ ] (cid:1) ∪ γ − n (cid:0) [ t, ∞ ] (cid:1) . Let U n be the component of ˜ C \ K n containing G ( T ) for sufficiently large T . If ε issufficiently small, then G ( t ) ∈ U n , hence A ⊂ U n and so A ⊂ \ n U n ⊂ G s ([ t, ∞ ))as desired. (Compare [R2, Corollary 11.4] for a similar proof, using parameter raysaccumulating on G s instead of the curves γ ± n ; recall Figure 2.) (cid:4) Lemma 13 can be applied, in particular, to the accumulation sets of parameter rays,as stated in the following corollary.
Corollary 14 (Accumulation sets of parameter rays) . Let G s be a parameter ray, andlet L denote the accumulation set of G s in ˜ C . If G is a parameter ray and G ( t ) ∈ L for some t ∈ (0 , ∞ ] , then G (cid:0) (0 , t ] (cid:1) ⊂ L .(In particular, the accumulation set of G is contained in L .) IFURCATION LOCI 13
Proof.
Apply the previous lemma to the sets A = A t = G s (cid:0) (0 , t ] (cid:1) with t >
0. Since A contains points at arbitrarily small potentials, and since parameter rays are pairwisedisjoint, the first alternative in the conclusion of the lemma cannot hold. Hence (writing f A t for the closure of A t in ˜ C as before) we have G (cid:0) (0 , t ] (cid:1) ⊂ \ t> f A t = L. (cid:4) We also require the following fundamental theorem from [RS1] about exponentialparameter space.
Theorem 15 (The Squeezing Lemma (connected sets version)) . Let A be an unboundedconnected subset of exponential parameter space which contains only finitely many in-different and no hyperbolic parameters, and let ˜ A be the closure of A in ˜ C . Then every s ∈ e A ∩ S is an infinite external address for which a parameter ray G s exists, and G s ( t ) ∈ e A for all sufficiently large t .Remark 1. In [RS1, Theorem 1.3], the Squeezing Lemma is formulated for curves ratherthan connected sets. The proof given there also proves the above version; compare thesketch below. The name of the result comes from the “squeezing” around a parameterray by nearby hyperbolic components (or parameter rays) as in the proof of Lemma 13.
Remark 2.
The idea of the proof of the Squeezing Lemma goes back to the originalproof [S1, Theorem V.6.5], [S3] that the boundary of every hyperbolic component inexponential parameter space is a connected subset of the plane. A suitable formulationof the Squeezing Lemma can also be used to prove this fact; see [RS1, Theorem 1.2].
Sketch of proof.
The proof requires a number of technical and combinatorial considera-tions. We will describe the underlying strategy and refer the reader to [RS1] for details.Let s ∈ e A ∩ S . First, let us show that s cannot be an intermediate external address.Indeed, otherwise there is a unique hyperbolic component W which is associated to s inthe following sense [S2]: if γ : [0 , ∞ ) → W is a curve along which the multiplier of theattracting orbit tends to zero, then s is the sole accumulation point of γ in W . Since A ∩ W is empty, we can draw a separation line through W and two child components(one slightly above W and one slightly below) which separates A from the address s .More details can be found in [RS1, Section 6].So s is an infinite external address; i.e. s ∈ Z N . If there is no parameter ray associatedto s , then this means [FS] that at least some of the entries in the sequence s growextremely fast. Every entry in s which exceeds all previous ones implies the existenceof a separation line in ˜ C which encloses parameter rays at external addresses near s ,and the domains thus enclosed shrink to s ∈ ˜ C . Hence some of these separation linesseparate A from external addresses near s , a contradiction. (The detailed proof uses thecombinatorial structure of internal addresses and can be found in [RS1, Section 7].) Itfollows that the address s must indeed have a parameter ray associated to it.We can hence apply Lemma 13 with t = ∞ . It follows that G s ( t ) ∈ ˜ A for allsufficiently large t , as claimed. (cid:4) We show below that every parameter ray has an accumulation point in B ∗ . For now,we only note the following. Lemma 16.
Every parameter ray has an accumulation point in B . (That is, a parameterray cannot land at infinity.)Proof. For the Mandelbrot set, this is clear. For exponential parameter space, it followsby applying the Squeezing Lemma to the connected set A := G s (cid:0) (0 , (cid:1) . Indeed, if A is bounded, then G s has at least one finite accumulation point, and there is nothing toprove. Otherwise, let ˜ A be the closure of A in ˜ C . Then there exists an infinite externaladdress s ′ ∈ ˜ A ∩ S so that G s ′ ( t ) ∈ ˜ A for all sufficiently large t . Since A does not containparameters on parameter rays at arbitrarily high potentials, it follows that G s ′ ( t ) belongsto A \ A , and hence to the accumulation set of G s . (cid:4) The following is essentially a weak version of Douady and Hubbard’s “branch theorem”[DH, Proposition XXII.3]; see also [S4, Theorem 3.1].
Lemma 17.
Let ˘ Y be a non-hyperbolic extended fiber. Then ˘ Y contains the accumulationset of at least one but at most finitely many parameter rays.Proof. For quadratic polynomials, the fact that every fiber contains the accumulation setof a parameter ray is immediate. Indeed, if c ∈ ∂ M , then c is contained in some primeend impression (compare e.g. [M, Chapter 17] or [P, Chapter 2] for background on thetheory of prime ends). This impression contains the accumulation set of an associatedparameter ray (this set is called the set of principal points of the prime end). Such anaccumulation set clearly cannot be separated from c by any separation line.That there are only finitely many such parameter rays follows from the usual BranchTheorem, see [DH, Expos´e XXII] or [S4, Theorem 3.1], which states that the Mandelbrotset branches only at hyperbolic components and at postsingularly finite parameters.For exponential maps, the finiteness statement follows from the corresponding factfor Multibrot sets. Indeed, suppose we had infinitely many parameter rays G s , G s , . . . which are not separated from one another by any separation line. Then it follows bycombinatorial considerations that all s j are bounded sequences, and they differ fromone another at most by 1 in each entry. (The argument is similar as in the proof ofthe Squeezing Lemma: if two addresses differ by more than 1 in one entry, then the“internal address algorithm” generates a separation line separating the two. Similarly,if s was unbounded, then for every bounded external address t there is a separation linewhich surrounds s — i.e., separates s from ∞ in ˜ C — and separates s from t . But anyseparation line which surrounds s must also surround all other s j , which contradicts thefact that there are bounded external addresses between any two elements of S .)So the entries of the addresses s j are uniformly bounded, and for every Multibrot set M d of sufficiently high degree d , there are parameter rays G s , G s , . . . at correspondingangles. The Branch Theorem for Multibrot sets [S4, Theorem 3.1 and Corollary 8.5]shows that some of the parameter rays G s i are separated from each other by a separationline for M d . But then it follows combinatorially that there is a similar separation linein exponential parameter space, in contradiction to our assumption. Compare [RS2,Theorem A.3].To see that there is at least one parameter ray contained in every non-hyperbolicextended fiber ˘ Y , recall from Lemma 12 that ˘ Y ∪ {∞} is a compact and connectedsubset of the Riemann sphere. So every component of ˘ Y is unbounded; the Squeezing IFURCATION LOCI 15
Lemma implies that each such component contains points on some parameter ray G s .But since separation lines cannot separate the ray G s , it follows that ˘ Y contains theentire ray G s and hence also its accumulation set. (cid:4) Lemma 18 (Extended fibers are connected) . Let ˘ Y be an extended fiber. Then ˘ Y and ˘ Y B := ˘ Y ∩ B are connected subsets of C .Proof. We can assume that ˘ Y is not a hyperbolic fiber, as otherwise the claim is trivial.For quadratic polynomials ˘ Y is a compact and connected subset of C (Lemma 12).Furthermore, every component of ˘ Y \ B (if any) would be a (non-hyperbolic) stablecomponent of the Mandelbrot set. Since all such components are simply connected,removing them from ˘ Y does not disconnect ˘ Y ∩ B .It remains to deal with the exponential case. Let G , . . . , G n be the parameter raysintersecting ˘ Y , with external addresses s , . . . , s n . Then all G i are contained in ˘ Y . Let˜ Y be the closure of ˘ Y in ˜ C ; then, by the Squeezing Lemma, ˜ Y = ˘ Y ∪ { s , . . . , s n } . Notethat ˜ Y is a compact connected subset of ˜ C (as in Lemma 12, it can be written as acountable nested intersection of compact connected subsets of ˜ C ).As in the quadratic setting, any stable component U of exponential maps is simplyconnected (since boundaries of hyperbolic components and parameter rays, both of whichare unbounded, are dense in B ). Hence ˜ Y B := ˘ Y B ∪ { s , . . . , s n } is also compact andconnected.Now suppose by contradiction that ˘ Y B = A ∪ A , where A and A are nonempty,closed and disjoint. Let ˜ A j be the closure of A j in ˜ C . Since ˜ Y B = ˜ A ∪ ˜ A is connected,we must have ˜ A ∩ ˜ A = ∅ . I.e., some s j belongs to both ˜ A and ˜ A ; let C be thecomponent of ˜ A containing s j .By the boundary bumping theorem [Na, Theorem 5.6], every component of ˜ A containsone of the addresses s i , so ˜ A has only finitely many connected components. This impliesthat C ) { s j } . By Lemma 13, it follows that C ∩ G j = ∅ . Since G j is connected, in fact G j ⊂ A . Likewise, G j ⊂ A , which contradicts the assumption that A ∩ A = ∅ .So ˘ Y B is connected, as is ˘ Y itself. (cid:4) Proof of Theorem 2. (Bifurcation locus is connected.)
Suppose that the bifurcation locus B is not connected. Then there is some stable component U such that two components C and C of ∂U belong to different connected components of C \ U . Since boundariesof hyperbolic components are connected subsets of C [S3, RS1] (recall Remark 2 afterTheorem 15), U must be a non-hyperbolic stable component, and hence U is contained ina single extended fiber ˘ Y (no separation line can separate any two points in U ). However,then C and C would belong to different components of ˘ Y \ U , which contradicts Lemma18. (cid:4) Remark.
Theorem 2 can also be proved directly from the “Squeezing Lemma”; see [RS1,Proof of Theorem 1.1].We have now proved a number of results regarding extended fibers. Fibers themselvescan be more difficult to deal with in the exponential setting, because they may (at least a priori ) not be closed. For example, from what we have shown so far, it is conceivable that an extended fiber is completely contained in I R , and hence does not contain a(reduced) fiber. We shall now show that this is not the case, and that any extendedfiber in fact intersects B ∗ in either one or uncountably many points. Theorem 19 (Accumulation sets of parameter rays) . Every parameter ray has at leastone accumulation point in B ∗ ; if there is more than one such point, there are in factuncountably many.Furthermore, let Y be a fiber. Then either • Y is trivial (i.e., consists of exactly one point); if Y is non-hyperbolic, then thereis at least one parameter ray landing at the unique point of Y ; or • Y is not trivial, in which case Y ∩ B ∗ is uncountable.Proof. For the quadratic family, the accumulation set of every parameter ray is containedin the boundary of the Mandelbrot set, which equals B ∗ . Any fiber Y is a closed,connected subset of M , and hence either consists of a single point or has the cardinalityof the continuum. If the fiber has interior, then its boundary is contained in B ∗ ∩ Y and also has the cardinality of the continuum. If the fiber is trivial, then clearly every parameter ray accumulating on Y must land at the single point of Y .So let us now consider the case of the exponential bifurcation locus. Let G s bea parameter ray, and let ˘ Y be the extended fiber containing G s . By Lemma 17, ˘ Y contains at most finitely many parameter rays G , . . . , G n , at addresses { s , . . . , s n } .(Our original ray G s will be one of these.) Denote by L i the set of all accumulationpoints of G i (as t →
0) in ˜ C .Note that, if L i ∩ I R = ∅ , say G j ( t ) ∈ L i for some j ∈ { , . . . , n } and t ∈ (0 , ∞ ], then G j (cid:0) (0 , t ] (cid:1) ⊂ L i and L j ⊂ L i by Corollary 14. Claim.
Let L ⊂ ( ˘ Y ∩ B ) ∪ { s , . . . s n } ⊂ ˜ C be nonempty, compact and connected, andsuppose that there are no i and t > with L ⊂ G i (cid:0) [ t, ∞ ] (cid:1) .Then L ∩ B ∗ is either uncountable or a singleton. In the latter case, (a) if L
6⊂ B ∗ , then there is j such that G j ∩ L = ∅ and G j lands at the unique pointof L ∩ B ∗ ; (b) every connected component of C \ L contains infinitely many parameter rays, andhence is not contained in ˘ Y . Note that, in (a), we do not claim that all rays which intersect L land at the uniquepoint of L ∩ B ∗ . To illustrate the claim in this case, it may be useful to imagine L to be the set from Figure 5. Other model cases to imagine are those where L is theunion of of finitely many parameter rays landing at a common point, or where L is anindecomposable continuum with one or more parameter rays dense in L (as e.g. in thestandard Knaster (or “buckethandle”) continuum [K, §
43, V, Example 1]).Using the claim, we can prove both statements of the theorem. For the first part, welet L be the accumulation set of G s ; by Corollary 14, we do not have L ⊂ G s (cid:0) [ t, ∞ ) (cid:1) forany i and t >
0, so the claim applies. For the second part, set L := ( ˘ Y ∩B ) ∪{ s , . . . , s n } .(The final part of the claim implies, in particular, that if L ∩ B ∗ = Y ∩ B ∗ is a singleton,then ˘ Y has no interior components, and hence Y is trivial, as claimed.) IFURCATION LOCI 17
Proof of the Claim.
Note that it follows from Lemma 13 and the assumption that, forevery j ∈ { , . . . , n } , there is some T ∈ [0 , ∞ ] such that L ∩ G j = G j (cid:0) (0 , T ] (cid:1) (where weunderstand the interval (0 , T ] to be empty in the case of T = 0).We will proceed by removing isolated end pieces of parameter rays from L . Moreprecisely, suppose that there is some j ∈ { , . . . , n } and some t > G j (cid:0) ( t, ∞ ] (cid:1) ∩ L is a relatively open subset of L . Choose t ≥ t > t have thisproperty.Then G j (cid:0) ( t , ∞ ] (cid:1) is a relatively open subset of L , which we will call an “isolated endpiece”. Note that the relative boundary of this piece in L is either the singleton G j ( t ),if t >
0, or the accumulation set L j of the ray G j otherwise; so in either case thisboundary is connected. By the boundary bumping theorem, L ′ := L \ G j (cid:0) ( t , ∞ ] (cid:1) isa compact connected subset of ˜ C . Furthermore, L j does not contain any point G j ( t )with t > t by choice of t . So L j ⊂ L ′ , and hence L ′ is nonempty and satisfies theassumptions of the claim. Note that L ′ ∩ B ∗ = L ∩ B ∗ .We can apply this observation repeatedly to remove such isolated end pieces of param-eter rays from L . Note that, if t = 0, then it could be that L ′ contains an isolated endpiece of a parameter ray which was not isolated in L . (Recall Figure 5.) However, sincethis happens at most n times, in finitely many steps we obtain a set L ⊂ L , satisfyingthe assumptions of the claim and with L ∩ B ∗ = L ∩ B ∗ , such that furthermore(*) if t > G j (cid:0) ( t, ∞ ] (cid:1) ∩ L = ∅ , then G j (cid:0) ( t, ∞ ] (cid:1) ∩ L is not relativelyopen in L .We observe that (*) implies the following stronger property:(**) if G j ( t ) ∈ L , then G j (cid:0) [ t, t ] (cid:1) is a nowhere dense subset of L for all t ∈ (0 , t ).Equivalently, there are no t ∈ (0 , t ) and ε < min( | t | , | t − t | ) such that I := G j (( t − ε, t + ε )) is relatively open in L . To prove (**), suppose by contradiction that such t and ε exist. By the boundary bumping theorem, every connected component of L \ I mustcontain one of the two endpoints of I . Hence there are at most two such components,and these are therefore both open and closed in L \ I . Furthermore, by Lemma 13,any component of L \ I which intersects G j (cid:0) [ t + ε, ∞ ] (cid:1) is contained in G j (cid:0) [ t + ε, ∞ ] (cid:1) .Together, these facts imply that G j (cid:0) ( t + ε, ∞ ] (cid:1) ∩ L is a relatively open subset of L \ I ,and hence of L , which contradicts (*).If L is a singleton, then L ⊂ B ∗ . Otherwise, L is a nondegenerate continuum, andin particular a complete metric space. Property (**) implies that L can be writtenas the union of L ∩ B ∗ with countably many nowhere dense subsets; if L ∩ B ∗ wascountable, this would violate the Baire category theorem.In the singleton case, write L = L ∩ B ∗ = { c } . Let I be the set of indices i with G i ∩ L = ∅ . Let us assume that I = ∅ , as otherwise there is nothing to prove. By reordering,we may also assume that I = { , . . . , k } , where 0 ≤ k ≤ n , and that furthermore G isthe last parameter ray which was completely removed in the construction of L , G isthe one completely removed before that, etc. By construction, the accumulation set L of G is contained in L , and hence G lands at c . In fact, we inductively get(1) L i ⊂ { c } ∪ i − [ j =1 G j . It follows that every component of ˜ C \ L contains an interval of S \ I , and hence infinitelymany parameter rays, as claimed.(One way of seeing this is to recall that ˜ C is homeomorphic to the unit disk in R . Itfollows from (1) and Janiszewski’s theorem (see [P, Page 2] or [Ne, Theorem V.9 · · L , considered as a subset of R in this manner, does not separate the plane. So everycomponent of ˜ C \ L must intersect the boundary of ˜ C in R , i.e. S , as required.) △ (cid:4) Proof of Theorem 7. (Properties of fibers.)
We just proved the fact that fibers are eithertrivial or uncountable. Also, we proved that every parameter ray has some accumulationpoint in B ∗ , and hence that every extended fiber intersects B ∗ . The fact that extendedfibers are connected was shown above in Lemma 18. (cid:4) Proof of Theorem 11. (Trivial fibers and landing of rays.)
Let Y be a fiber, and supposethat every point of Y is the landing point of a parameter ray. By Lemma 17, this meansthat Y is finite. Hence, by Theorem 19, Y is trivial.The converse follows directly from Theorem 19. (cid:4) Finally, let us prove the two remaining theorems, which deal exclusively with theMandelbrot set M . Proof of Theorem 10. (Trivial fibers and local connectivity.)
It is easy to see that trivi-ality of a fiber Y in the Mandelbrot set implies local connectivity of M at Y . Indeed, asnoted above, Y can be written as the nested intersection of countably many connectedclosed subsets of B , each of which is a neighborhood of Y . (Compare [S4, Proposi-tion 4.5].)For the converse direction, suppose that M is locally connected at every point of Y .Let z ∈ Y be an accumulation point of some parameter ray G . Then there is a sequence C k of cross-cuts of the domain W := ˆ C \ M (i.e. C k is a Jordan arc intersecting M onlyin its two endpoints) with the following properties. • C k separates ∞ from all points on G with sufficiently small potential. • The arcs C k converge to { z } in the Hausdorff distance.Let W k be the component of W \ C k not containing ∞ . Then I G := T k W k is the prime end impression of the parameter ray G . We will show that I = { z } .Indeed, let ε >
0. Since M is locally connected at z , we can find a connectedneighborhood K of z in M of diameter less than ε , for any ε >
0. Since closures ofconnected sets are connected, we may assume that K is closed. For sufficiently large k , the arc C k is a crosscut of K , and is contained in the disk of radius ε around z . Itfollows that diam W k ≤ ε , and hence diam I ≤ ε . Since ε > I = { z } , as claimed. The boundary of any fiber Y is contained in the union of the prime end impressionscorresponding to the parameter rays accumulating at Y . By Lemma 17, there are only In the terminology of prime ends, we have just shown that the complement of a simply connecteddomain cannot be locally connected at any principal point of a nontrivial prime end impression. Evenmore is true: a prime end impression can contain at most two points in which the complement of thedomain is locally connected; compare [R3].
IFURCATION LOCI 19 finitely many such parameter rays. As we have just shown, for each of these rays theprime end impressions consist of a single point. The boundary of fiber Y is thus finite.Since Y is connected, it follows that Y is trivial as claimed. (cid:4) Proof of Theorem 4. (MLC implies density of hyperbolicity.)
By Theorem 10, local con-nectivity of the Mandelbrot set is equivalent to triviality of fibers; by Theorem 9, trivi-ality of fibers implies density of hyperbolicity. (cid:4)
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