Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps
PPOSITIVE TRANSVERSALITY VIA TRANSFER OPERATORS ANDHOLOMORPHIC MOTIONS WITH APPLICATIONS TO MONOTONICITYFOR INTERVAL MAPS
GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN
Abstract.
In this paper we will develop a general approach which shows that generalized“critical relations” of families of locally defined holomorphic maps on the complex plane unfoldtransversally. The main idea is to define a transfer operator, which is a local analogue of theThurston pullback operator, using holomorphic motions. Assuming a so-called lifting propertyis satisfied, we obtain information about the spectrum of this transfer operator and thus abouttransversality. An important new feature of our method is that it is not global: the maps weconsider are only required to be defined and holomorphic on a neighbourhood of some finite set.We will illustrate this method by obtaining transversality for a wide class of one-parameterfamilies of interval and circle maps, for example for maps with flat critical points, but also formaps with complex analytic extensions such as certain polynomial-like maps. As in Tsujii’sapproach [48, 49], for real maps we obtain positive transversality (where > holds instead ofjust (cid:54) = 0 ), and thus monotonicity of entropy for these families, and also (as an easy application)for the real quadratic family.This method additionally gives results for unimodal families of the form x (cid:55)→ | x | (cid:96) + c for (cid:96) > not necessarily an even integer and c real. Contents
1. Introduction 12. Statement of Results 33. Organisation of this paper and outline of the proof 94. The spectrum of a transfer operator A and transversality 95. The lifting property and the spectrum of A f λ ( x ) = f ( x ) + λ and f λ ( x ) = λf ( x ) f c ( x ) = | x | (cid:96) ± + c with (cid:96) ± > large 26Appendix B. Families without the lifting property 31Appendix C. The lifting property in the setting of rational mappings 32References 331. Introduction
This paper is about bifurcations in families of (real and complex) one dimensional dynamicalsystems. For example, for real one-dimensional dynamical systems, we have a precise combina-torial description on the dynamics in terms of the so-called kneading sequences. One simple butvery important question is how the kneading sequence varies in families of such systems. Forthe real quadratic family f a ( x ) = x + a , it is known that the kneading sequence depends mono-tonically on the parameter a (with respect to the natural order defined for kneading sequences).Interestingly the proofs of this result, by Milnor-Thurston, Douady-Hubbard and Sullivan, make Date : 24 Jan 2019. This paper is based on the preprint
Monotonicity of entropy and positively oriented transversality for familiesof interval maps , see https://arxiv.org/abs/1611.10056 a r X i v : . [ m a t h . D S ] M a r GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN use of Teichmüller theory, uniqueness in Thurston’s realisation theorem, or quasiconformal rigid-ity theory and that the map f a is quadratic, see [37, 47] and also [11].To answer the above monotonicity question it is enough to show that when f na (cid:48) (0) = 0 forsome a (cid:48) ∈ R , then there exists no other parameter a (cid:48)(cid:48) ∈ R for which f na (cid:48)(cid:48) (0) = 0 and for which f a (cid:48) and f a (cid:48)(cid:48) also have the same (periodic) kneading sequence.If dda f na (0) | a = a (cid:48) (cid:54) = 0 (which is called transversality ) then one has local uniqueness in thefollowing sense: there exists (cid:15) > so that the kneading invariant of f a for a ∈ ( a (cid:48) − (cid:15), a (cid:48) ) ,for a ∈ ( a (cid:48) , a (cid:48) + (cid:15) ) and for a = a (cid:48) are all different. It turns out that global uniqueness andmonotonicity follows from Q := dda f na (0) | a = a (cid:48) Df na (cid:48) ( f a (cid:48) (0)) > (which we call positive transversality ). Tsujii gave an alternative proof of the above monotonicityfor the quadratic family by showing that this inequality holds [48, 49].For general (complex) holomorphic families of maps with several critical points which all areeventually periodic there exists a similar expression Q . Again Q (cid:54) = 0 implies that the bifurcationsare non-degenerate and hence the corresponding critical relations unfold transversally. In thispaper we show that the inequality Q (cid:54) = 0 holds provided the spectrum of some operator A doesnot contain 1, and that Q > holds if additionally the spectrum of A is contained in the closedunit disc and the family of maps is real.We define this operator A by considering how the speed of a (holomorphic) motion of theorbits of critical points is lifted under the dynamics. The novelty of our method, describedin Proposition 5.1 and Theorem 6.1, is to show if these holomorphic motions have the liftingproperty, i.e. can be lifted infinitely many times over the same domain, then the operator A hasthe above spectral properties.It turns out that the lifting property makes minimal use of the global dynamics of the holo-morphic extension of the dynamics. Thus we can obtain transversality properties of families f t of maps defined on open subsets of the complex plane so that f has a finite invariant markedset, e.g., f is ‘critically finite’.The methods developed in this paper give a new and simple proof of well-known results forfamilies of polynomial maps, rational or entire maps, but also applies to many other families forwhich no techniques were available. For example we obtain monotonicity for the family of maps f c ( x ) = be − / | x | (cid:96) + c having a flat critical point at . We also obtain partial monotonicity for the family f c ( x ) = | x | (cid:96) + c when (cid:96) is large.As mentioned, the aim of this paper is to deal with families of maps which are only locallyholomorphic. This means that the approach pioneered by Thurston, and developed by Douadyand Hubbard in [13], cannot be applied. In Thurston’s approach, when f is a globally definedholomorphic map, P is a finite f -forward invariant set containing the postcritical set and theThurston map σ f : Teich ( (cid:98) C \ P ) → Teich ( (cid:98) C \ P ) is defined by pulling back an almost holomorphicstructure. It turns out that σ f is contracting, see [13, Corollary 3.4]. In Thurston’s result onthe topological realisation of rational maps, Douady & Hubbard [13] use that the dual of thederivative of the Thurston map σ f is equal to the Thurston pushforward operator f ∗ .The pushforward operator acts on space of quadratic differentials, and the injectivity of f ∗ − id has been used to obtain transversality results for rational maps or even more general spaces ofmaps, see Tsujii [48, 49], Epstein [16], Levin [28, 29, 30], Makienko [32], Buff & Epstein [6] andAstorg [1]. See also earlier [31], [14, Proposition 22 ], [15]. For a short and elementary proof thatcritical relations unfold transversally in the setting of rational maps, see [22]. For the spectrumof f ∗ see also [25], [26], [27], [18] and [7]. In [45], transversality is shown for rational maps for This reference was provided by the author. which each critical point is mapped into a hyperbolic set, using the uniqueness part of Thurston’srealisation result.However if for example f : U → V is a polynomial-like map then each point in the boundaryof V is a singular value, Teich ( V \ P ) is infinite dimensional, Thurston’s algorithm is only locallywell-defined and it is not clear whether it is locally contracting.The purpose of our paper is to bypass this issue, by going back to the original Milnor-Thurstonapproach. Milnor and Thurston [37] associated to the space of quadratic maps and the combi-natorial information of a periodic orbit, a map which assigns to a q tuple of points a new q tupleof points, F : ( z , ...z q ) (cid:55)→ (ˆ z , ..., ˆ z q ) where ˆ z q = 0 and f z (ˆ z i ) = z i +1 mod q where f c ( z ) ≡ z + c . Since F is many-valued, Milnor & Thurston considered a lift ˜ F of thismap to the universal cover and apply Teichmüller theory to show that ˜ F is strictly contractingin the Teichmuller metric of the universal cover.We bypass this issue by rephrasing their approach locally (via holomorphic motions). Thisis done in the set-up of so-called marked maps (and their local deformations) which includeparticularly critically finite maps with any number of critical (singular) points. In the first partof the paper we prove general results, notably the Main Theorem, which show that under theassumption that some lifting property holds for the deformation, either some critical relationpersists along some non-trivial manifold in parameter space or one has transversality, i.e. thecritical relation unfolds transversally. Here the lifting property is an assumption that sequencesof successive lifts of holomorphic motions are compact. In the second part of the paper, we thenshow that this lifting property holds not only in previously considered global cases but also forinteresting classes of maps where the ‘pushforward’ approach breaks down.More precisely, we define a transfer operator A by its action on infinitesimal holomorphicmotions on P . It turns out that if the lifting property holds, then the spectrum of the operator A lies inside the unit disc. Moreover, if the operator A has no eigenvalue then transversalityholds; in the real case one has even positive transversality (the sign of some determinant ispositive). One of the main steps in the proof of the Main Theorem is then to show that ifthe operator has an eigenvalue then the critical relation persists along a non-trivial manifoldin parameter space. It turns out that for globally defined critically finite maps f the transferoperator A can be identified with (dual of) f ∗ .By verifying the lifting property we recover previous results such as transversality for rationalmaps, but also obtain transversality for many interesting families of polynomial-like mappingsand families of maps with essential singularities. For real local maps our approach gives the‘positive transversality’ condition which first appeared in [48, 49] and therefore monotonicity ofentropy for certain families of real maps.2. Statement of Results
The notion of a marked map. A marked map is a map g from the union of a finite set P and an open set U ⊂ C mapping into C such that • there exists a finite set P ⊃ P such that g ( P ) ⊂ P and P \ P ⊂ U ; • g | U is holomorphic and g (cid:48) ( x ) (cid:54) = 0 for x ∈ P \ P .Let { c , , . . . , c ,ν } denote the (distinct) points in P and write c = c ( g ) = ( c , , . . . , c ,ν ) and c = c ( g ) = ( g ( c , ) , . . . , g ( c ,ν )) := ( c , , c , , . . . , c ,ν ) . Remark 2.1. So P is a forward invariant set for g , and g is only required to be holomorphic(and defined) on a neighbourhood of P \ P . A marked map g does not need to be defined in aneighbourhood of P . In applications, points in P will be where some extension of g has a criticalpoint, an (essential) singularity or even where g has a discontinuity. In this sense marked mapscorrespond to a generalisation of the notion of critically finite maps. GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN
Holomorphic deformations. A local holomorphic deformation of g is a triple ( g, G, p ) W with the following properties:(1) W is an open connected subset of C ν containing c ( g ) ;(2) p = ( p , p , . . . , p ν ) : W → C ν is a holomorphic map, so that p ( c ) = c ( g ) (and so allcoordinates of p ( c ) are distinct).(3) G : ( w, z ) ∈ W × U (cid:55)→ G w ( z ) ∈ C is a holomorphic map such that G c = g | U . Example 2.2. (i) The simplest local holomorphic deformation of g is of course the trivial one: G w ( z ) = g ( z ) , p ( w ) = c , ∀ w .(ii) If g is a marked map with ν = 1 , i.e., P = { c } , then G w ( z ) = g ( z ) + ( w − g ( c )) , p ( w ) ≡ c ,defines a local holomorphic deformation of g .(iii) If g is rational map with ν critical points c , . . . , c ν with multiplicity µ , . . . , µ ν then thereexists a local holomorphic deformation ( g, G, p ) W of g so that w i = G w ( p i ( w )) is a critical valueof G w for each w = ( w , . . . , w ν ) ∈ W and each i = 1 , . . . , ν , see Appendix C. Transversal unfolding of critical relations.
Let us fix ( g, G, p ) W as above. Since g ( P ) ⊂ P and P is a finite set, for each j = 1 , , . . . , ν , exactly one of the following criticalrelations holds:(a) There exists an integer q j > and µ ( j ) ∈ { , , . . . , ν } such that g q j ( c ,j ) = c ,µ ( j ) and g k ( c ,j ) (cid:54)∈ P for each ≤ k < q j ;(b) There exist integers ≤ l j < q j such that g q j ( c ,j ) = g l j ( c ,j ) and g k ( c ,j ) (cid:54)∈ P for all ≤ k ≤ q j .Relabelling these points c ,j , we assume that there is r such that the first alternative happensfor all ≤ j ≤ r and the second alternative happens for r < j ≤ ν .Define the map R = ( R , R , . . . , R ν ) from a neighbourhood of c ∈ C ν into C ν as follows: for ≤ j ≤ r ,(2.1) R j ( w ) = G q j − w ( w j ) − p µ ( j ) ( w ) and for r < j ≤ ν ,(2.2) R j ( w ) = G q j − w ( w j ) − G l j − w ( w j ) , where w = ( w j ) νj =1 . Definition 2.3.
We say that the holomorphic deformation ( g, G, p ) W of g satisfies the transver-sality property , if the Jacobian matrix D R ( c ) is invertible. Example 2.4. (i) Assume that ( g, G, p ) W is a local holomorphic deformation so that for each w = ( w , . . . , w ν ) ∈ W , the critical values of G w are w , . . . , w ν and p ( w ) , . . . , p ν ( w ) are thecritical points of G w . Then G w ( p i ( w )) = w i . Hence R j ≡ in (2.1) or (2.2) correspond to G q j w ( p j ( w )) − p µ ( j ) ( w ) ≡ resp. G q j w ( p j ( w )) − G l j w ( p i ( w )) ≡ . These equations define the set of parameters w for which the corresponding ‘critical relation’ issatisfied within the family G w .(ii) In Example 2.6(ii) we will consider a holomorphic deformation ( g, G, p ) W of a map g sothat G w is not defined (as an analytic map) in p j ( w ) , but nevertheless the above interpretationis valid.(iii) If we take the trivial deformation G w ( z ) = g ( z ) , p ( w ) = c , ∀ w ∈ W , then defini-tions (2.1) or (2.2) take the form R j ( w ) = g q j − ( w j ) − c ,j (cid:48) respectively R j ( w ) = g q j − ( w j ) − g l j − ( w j ) . Then D R ( c ) is a diagonal matrix with entries Dg q j − ( w j ) for ≤ j ≤ r and Dg l j − ( w j )[ Dg q j − l j ( g l j − ( w j )) − for r < j ≤ ν . So the matrix D R ( c ) is non-degenerate iff Dg q j − l j ( g l j − ( w j )) (cid:54) = 1 for r < j ≤ ν . It follows immediately that the holomorphic deformation ( g, G, p ) W satisfies the transversality property if and only if Dg q j − l j ( g l j ( c ,j )) (cid:54) = 1 for r < j ≤ ν .We should emphasise that in this setting the condition R ( w ) = 0 has nothing to do with thepresence of critical relations. Real marked maps and positive transversality.
A marked map g is called real if P ⊂ R and for any z ∈ U we have z ∈ U and g ( z ) = g ( z ) . Similarly, a local holomorphic deformation ( g, G, p ) W of a real marked map g is called real if for any w = ( w , w , . . . , w ν ) ∈ W , z ∈ U and j = 1 , , . . . , ν , we have w = ( w , w , . . . , w ν ) ∈ W , G w ( z ) = G w ( z ) , and p j ( w ) = p j ( w ) . Definition 2.5.
Let ( g, G, p ) W be a real local holomorphic deformation of a real marked map g .We say that the unfolding ( g, G, p ) W satisfies the ‘positively oriented’ transversality property if (2.3) Q := det( D R ( c )) (cid:81) νj =1 Dg q j − ( c ,j ) > . The sign in the previous inequality means that the intersection of the analytic sets R j = 0 j = 1 , . . . , ν , is not only in general position (i.e. ‘transversal’), but that the intersection patternis everywhere ‘positively oriented’ .2.5. The lifting property.
Let X ⊂ C and Λ be a domain in C which contains . As usual,we say that h λ is a holomorphic motion of X over (Λ , , if h λ : X → C satisfies:(i) h ( x ) = x , for all x ∈ X ,(ii) h λ ( x ) (cid:54) = h λ ( y ) whenever x (cid:54) = y and λ ∈ Λ , x, y ∈ X and(iii) Λ (cid:51) λ (cid:55)→ h λ ( z ) is holomorphic.Given a holomorphic motion h λ of g ( P ) over (Λ , , where Λ is a domain in C which contains , we say that ˆ h λ is a lift of h λ over Λ ⊂ Λ (with ∈ Λ ) with respect to ( g, G, p ) W if for all λ ∈ Λ , • ˆ h λ ( c ,j ) = p j ( c ( λ )) for each j = 1 , , . . . , ν , with c ,j ∈ g ( P ) , where c ( λ ) = ( h λ ( c , ) , h λ ( c , ) , . . . , h λ ( c ,ν )); • G c ( λ ) (ˆ h λ ( x )) = h λ ( g ( x )) for each x ∈ g ( P ) \ P .Clearly such a lift exists, provided Λ is contained in a sufficiently small neighbourhood of .We say that the triple ( g, G, p ) W has the lifting property if for each holomorphic motion h (0) λ of g ( P ) over ( D , there exist ε > and a sequence of holomorphic motions h ( k ) λ , k = 1 , , , . . . of g ( P ) over ( D ε , such that for each k ≥ , • h ( k +1) λ is a lift of h ( k ) λ over ( D ε , ; • there exists M > such that | h ( k ) λ ( x ) | ≤ M for all x ∈ g ( P ) and all λ ∈ D ε .In the case ( g, G, p ) W is real, we say it has the real lifting property if the corresponding propertyholds for any real-symmetric holomorphic motions h (0) λ .2.6. Statement of the Main Theorem.Main Theorem.
Assume that g does not have a parabolic periodic point in P \ P and that ( g, G, p ) W satisfies the lifting property. Then exactly one of the following holds:(1) the holomorphic deformation ( g, G, p ) W of g satisfies the tranversality property;(2) there exists a neighborhood W (cid:48) of c such that { w ∈ W (cid:48) : R ( w ) = 0 } is a smooth complexmanifold of positive dimension.Moreover, if ( g, G, p ) W is real and satisfies the real lifting property then in (1) ‘the transversalityproperty’ can be replaced by ‘the ‘positively oriented’ transversality property’. The statement of this theorem is a combination of the more detailed statements in Theo-rems 4.1 and 6.1.2.7.
Classical settings where the lifting property holds.
In many cases it is easy to checkthat the lifting property holds, and therefore the previous theorem applies. Indeed, it is easy tosee that this holds in the setting of polynomial or rational maps, see Section C.
GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN
Transversality for new families of maps corresponding to classes F , E , E o . In thissubsection we will discuss two new settings where the current approach can be applied to obtaintransversality.Let us first consider families of maps f c ( z ) = f ( z ) + c . Here f is contained in the space F ofholomorphic maps f : U → V where(a) U is a bounded open set in C such that ∈ U ;(b) V is a bounded open set in C ;(c) f : U \ { } → V \ { } is an unbranched covering;(d) V ⊃ B (0; diam( U )) ⊃ U .Examples of such families are Example 2.6. (i) f c ( z ) = z d + c , where U, V are suitably large balls and c ∈ U .(ii) f c ( x ) = be − / | x | (cid:96) + c where (cid:96) ≥ , b > e(cid:96) ) /(cid:96) , c ∈ U . Here U = U − ∪ U + , U + = − U − are topological disks symmetric w.r.t. the real axis and V is a punctured disc. That f ∈ F is proved in Corollary 7.3. Theorem 2.1.
Let f ∈ F , let g = f c and for each w ∈ W := C define G w ( z ) = g ( z ) + ( w − c ) and p ( w ) = 0 where we assume that c ∈ U . Moreover, assume that there exists q so that c n = g n − ( c ) ∈ U for all n < q and either c q = 0 or c q ∈ { c , . . . , c q − } . Moreover, assume c n / ∈ { , c , . . . , c n − } for < n < q . Then • ( g, G, p ) W satisfies the lifting property and transversality holds. • if ( g, G, p ) W is real, then positive transversality holds. Our methods also apply to families of the form f w ( z ) = wf ( z ) where f is contained in thespaces E and E o defined as follows. Consider holomorphic maps f : D → V such that:(a) D, V are open sets which are symmetric w.r.t. the real line so that f ( D ) = V (b) Let I = D ∩ R then there exists c > so that I ∪ { c } is a (finite or infinite) openinterval and ∈ I , c ∈ int ( I ) . Moreover, f extends continuously to I , f ( I ) ⊂ R and lim z ∈ D,z → f ( z ) = 0 .(c) Let D + be the component of D which contains I ∩ ( c, ∞ ) , where D + might be equal to D . Then u ∈ D \ { } and v ∈ D + \ { } , v (cid:54) = u , implies u/v ∈ V .Let E be the class of maps which satisfy ( a ) , ( b ) , ( c ) and assumption ( d ) :(d) f : D → V has no singular values in V \ { , } and c > is minimal such that f has apositive local maximum at c and f ( c ) = 1 .Similarly let E o be the class of maps which satisfy ( a ) , ( b ) , ( c ) and assumption ( e ) :(e) f is odd, f : D → V has no singular values in V \ { , ± } and c > is minimal such that f has a positive local maximum at c and f ( c ) = 1 .Here, as usual, we say that v ∈ C is a singular value of a holomorphic map f : D → C if it is acritical value, or an asymptotic value where the latter means the existence of a path γ : [0 , → D so that γ ( t ) → ∂D and f ( γ ( t )) → v as t ↑ . Note that we do not require here that V ⊃ D .Classes E and E o are rich even in the case D = C . See [19] for a general method of constructingentire (or meromorphic) functions with prescribed asymptotic and critical values. These classesare also non-empty when V = C and the domain D is a topological disk or even if D notsimply-connected [17]. V can also be a bounded subset of C , see example (v) below.Concrete examples of functions f of the class E are, where in (i)-(iv) we have D = V = C ,(i) f ( z ) = 4 z (1 − z ) ,(ii) f ( z ) = 4 exp( z )(1 − exp( z )) ,(iii) f ( z ) = [sin( z )] ,(iv) f ( z ) = m − m ( ez ) m exp( − z ) when m is a positive even integer.(v) the unimodal map f : [0 , → [0 , defined by f ( x ) = exp(2 (cid:96) ) (cid:16) − exp( − / | x − / | (cid:96) ) + exp( − (cid:96) ) (cid:17) satisfies f (0) = f (1) = 0 , f (1 /
2) = 1 and has a flat critical point at c = 1 / ; this maphas an extension f : D → V which is in E and for which V a punctured bounded disc provided (cid:96) is big enough and D = D − ∪ D + are disjoint open topological discs so that D − ∩ R = (0 , / and D + ∩ R = (1 / , , see Lemma 7.5.Examples of maps in the class E o are(vi) f ( z ) = sin( z ) and(vii) f ( z ) = ( m/ − m/ e m/ z m exp( − z ) when m is a positive odd integer. Theorem 2.2.
Let f ∈ E ∪ E o and for each w ∈ W := D + define G w ( z ) = w · f ( z ) and p ( w ) = c .Take c ∈ D , g = G c and assume that there exists q so that c n = g n − ( c ) ∈ D for all n ≤ q and either c q = c or c q ∈ { c , . . . , c q − } . Moreover, assume c n / ∈ { c , c , . . . , c n − } for < n < q .Then • ( g, G, p ) W satisfies the lifting property and transversality holds. • if ( g, G, p ) W is real, then positive transversality holds. Applications to monotonicity of topological entropy of interval maps.Corollary 2.7.
Take f ∈ F and consider the family f c = f + c , c ∈ J = U ∩ R . Then thekneading sequence K ( f c ) is monotone increasing in c ∈ J . Moreover, whenever c ∗ ∈ J is so that f qc ∗ (0) = 0 and f kc ∗ (0) (cid:54) = 0 for all ≤ k < q the following positive transversality condition (2.4) Q = ddc f qc (0) | c = c ∗ Df q − c ∗ ( c ∗ ) > . holds and the topological entropy of f c is decreasing in c ∈ J .The same statement holds for f c = c · f for f ∈ E ∪ E , except in this case we consider thetopological entropy of the unimodal map f | (0 , b ) where b = sup { b (cid:48) ∈ I : b (cid:48) > , f ( y ) > ∀ y ∈ (0 , b (cid:48) ) } . Monotonicity of entropy was proved in the case f c ( x ) = x + c in the 1980s as a major resultin unimodal dynamics. By now there are several proofs, see [37, 47, 11, 48, 49]. All these proofsuse complex analytic methods and rely on the fact that f c extends to a holomorphic map on thecomplex plane. These methods work well for f c ( x ) = | x | (cid:96) + c when (cid:96) is a positive even integerbut break down for general (cid:96) > and also for other families of non-analytic unimodal maps. Noapproach using purely real-analytic method has so far been successful in proving monotonicityfor any (cid:96) > . The approach to prove monotonicity via the inequality (2.4) was also previouslyused by Tsujii [48, 49] for real maps of the form z (cid:55)→ z + c , c ∈ R . Remark 2.8.
Let U denote the collection of unimodal maps f : R → R which are strictlydecreasing in ( −∞ , and strictly increasing in [0 , ∞ ) . The Milnor-Thurston kneading sequence of f ∈ U is defined as a word K ( f ) = i i · · · ∈ { , , − } Z + , where i k = if f k (0) > if f k (0) = 0 − if f k (0) < . For g ∈ U with K ( g ) = j j · · · , we say that K ( f ) ≺ K ( g ) if there is some n ≥ such that i k = j k for all ≤ k < n and (cid:81) nk =1 i k < (cid:81) nk =1 j k . Remark 2.9 ( Positive transversality and topological entropy).
Because f has a minimumat , x (cid:55)→ f qc ∗ ( x ) has a local maximum (minimum) at if Df q − c ∗ ( f c ∗ (0)) < (resp. > ). HenceEquation (2.4) implies that if has (precisely) period q at some parameter c ∗ , then ddc f qc (0) (cid:12)(cid:12) c = c ∗ < if f qc ∗ has a local maximum at , ddc f qc (0) (cid:12)(cid:12) c = c ∗ > if f qc ∗ has a local minimum at . Hence the number of laps of f nc (and therefore the topological entropy) is non-increasing when c increases. These inequalities also show that the multiplier λ ( c ) of the (local) analytic contin-uation p ( c ) of this periodic point of period q is strictly increasing. Note that there is a resultof Douady-Hubbard which asserts that in each hyperbolic component of the family of quadraticmaps, the multiplier of the periodic attractor is a univalent function of the parameter. Proving GENADI LEVIN, WEIXIAO SHEN AND SEBASTIAN VAN STRIEN (2.4) complements this by also showing that on the real line the multiplier of the periodic pointis increasing.
When f c ( x ) = | x | (cid:96) + c , and (cid:96) is not an integer, we have not been able able to prove the liftingproperty. The next theorem, which will be proved in Appendix A, gives monotonicity when (cid:96) isa large real number (not necessarily an integer), but only if not too many points in the criticalorbit are in the orientation reversing branch. Theorem A.1.
Let (cid:96) − , (cid:96) + ≥ and consider the family of unimodal maps f c = f c,(cid:96) − ,(cid:96) + where f c ( x ) = (cid:26) | x | (cid:96) − + c if x ≤ | x | (cid:96) + + c if x ≥ . For any integer L ≥ there exists (cid:96) > so that for any q ≥ and any periodic sequence i = i i · · · ∈ {− , , } Z + of period q so that (2.5) { ≤ j < q ; i j = − } ≤ L, and any pair (cid:96) − , (cid:96) + ≥ (cid:96) there is at most one c ∈ R for which the kneading sequence of f c isequal to i . Moreover, if i is realisable (i.e. if c = c ∗ exists) and i has minimal period q thenpositive transversality holds (2.6) Q = ddc f qc (0) | c = c ∗ Df q − c ∗ ( c ∗ ) > . The proof of this theorem uses delicate geometric arguments, see Appendix A. Note thatthere is an elegant algebraic proof of transversality for critically finite quadratic polynomials in[12, Chapter 19]. This proof also works for x (cid:55)→ | x | n +1 + c provided n is a positive integer, butit does not give the sign, so no monotonicity for this family can be deduced.2.10. Monotonicity along curves with one free critical point.
The above results requirethat all critical points are eventually periodic. Nevertheless, they also give information aboutthe bifurcations that occur for example along a curve L ∗ in parameter space corresponding to ( ν − -critical relations. The results in Section 8 informally state: Informal Statement of Theorem 8.1.
Critical relations unfold everywhere in the same direc-tion along L ∗ . This makes it possible to obtain information about monotonicity of entropy along the bonecurves considered in [38, Figure 11] and [40, Figure 8]. Indeed we obtain an alternative proof forone of the main technical steps in [38] in Theorem 8.2. Could such a simplification be made inthe case with at least three critical points?Indeed, it would be interesting to know whether the sign in (2.3) makes it possible to simplifythe existing proofs of Milnor’s conjecture. This conjecture is about the space of real polynomialswith only real critical points, all of which non-degenerate, and asks whether the level sets ofconstant topological entropy are connected. The proof of this conjecture in [38] in the cubic caseand in [5] for the general case relies on quasi-symmetric rigidity, but does having a positive signin (2.3) everywhere allow for a simplification of the proof of this conjecture?2.11.
Other applications.
Our approach can also be applied to many other settings, such asfamilies of Arnol’d maps, families of piecewise linear maps and to families of intervals maps withdiscontinuities (i.e. Lorenz maps), see [23, 24].Even though we deal with the polynomial and rational case in Appendix C, since it is soimportant, in a separate paper [22] we have given a very elementary proof of transversalityand related results in that setting, but without the sign in (2.6) and (2.3). In that paper thepostcritical set is allowed to be infinite . See [16] for an alternative discussion on transversalityfor maps of finite type, and [6] when the postcritical set is finite.
Acknowledgment.
We are indebted to Alex Eremenko for very helpful discussions concerningSubsection 7.3. The first author acknowledges the support of ISF grant 1226/17 grant, the secondauthor acknowledges the support of NSFC grant no: 11731003, and the last author acknowledges the support of ERC AdG grant no: 339523 RGDD. We would also like to thank the referee forsome very useful suggestions.3.
Organisation of this paper and outline of the proof
In this paper we consider holomorphic maps g : U → C where U is an open subset of thecomplex plane, together with a finite forward invariant marked set P , for example the postcriticalset. These maps do not necessarily have to be rational or transcendental. The aim is to show thatcritical relations of such a marked map unfold transversally under a holomorphic deformation G of g . We do this as follows. First, in Section 4, we associate a linear operator A : C g ( P ) → C g ( P ) by the action of G induced by lifting holomorphic motions on g ( P ) and show / ∈ spec ( A ) ⇔ transversality , spec ( A ) ⊂ D \ { } and G real = ⇒ positive transversality . More precisely, it is shown in Theorem 4.1 that the dimension of kernel of D R ( c ) is equal tothe geometric multiplicity of the eigenvalue of A . In Section 5 we then showlifting property = ⇒ spec ( A ) ⊂ D Then in Section 6 we show that provided the lifting property holds, { w ; R ( w ) = 0 } is locallya smooth submanifold whose dimension is equal to the geometric multiplicity of the eigenvalue of A . In applications, it is usually quite easy to show that the parameter set { w ; R ( w ) = 0 } cannot be a manifold of dimension > , and therefore that (cid:54)∈ spec ( A ) and so transversalityholds.It follows that transversality essentially follows from the lifting property. In Section C we showthat the lifting property holds in some classical settings. In Sections 7 we will show the liftingproperty holds for polynomial-like mappings from a separation property, and for maps from theclasses E , E o . In this way, we derive transversality for many families of interval maps, for examplefor a wide class of one-parameter families of the form f λ ( x ) = f ( x ) + λ and f λ ( x ) = λf ( x ) . As aneasy application, we will recover known transversality results for the family of quadratic maps,and address some conjectures from the 1980’s about families of interval maps of this type.In Appendix A we will study the family x (cid:55)→ | x | (cid:96) + c . When (cid:96) is not an even integer, we havenot been able to prove the lifting property in general. Nevertheless we will obtain the liftingproperty under additional assumptions.In Appendix B we give some examples for both transversality and the lifting property fails tohold.In a companion paper we show that the methods developed in this paper also apply to otherfamilies, including some for which separation property does not hold, such as the Arnol’d family.We also obtain positively oriented transversality for piecewise linear interval maps and intervalmaps with discontinuities (i.e. Lorenz maps), see also [23]4. The spectrum of a transfer operator A and transversality In this section we define a transfer operator A associated to the analytic deformation ofa marked map, and show that if is not an eigenvalue of A then transversality holds. Ifthe spectrum of A is inside the closed unit circle, we will obtain additional information abouttransversality, see Section 4.3.4.1. A transfer operator associated to a deformation of a marked map.
In §2.5, wedefined lift of holomorphic motions of g ( P ) associated to ( g, G, p ) W . Obviously there is a linearmap A : C g ( P ) → C g ( P ) such that whenever ˆ h λ is a lift of h λ , we have A (cid:32)(cid:26) ddλ h λ ( x ) | λ =0 (cid:27) x ∈ g ( P ) (cid:33) = (cid:26) ddλ ˆ h λ ( x ) | λ =0 (cid:27) x ∈ g ( P ) . We will call A the transfer operator associated to the holomorphic deformation ( g, G, p ) W of g . If both g and ( g, G, p ) W are real, then A ( R ν ) ⊂ R ν . In this case, we shall often consider realholomorphic motions, i.e. Λ (cid:51) is symmetric with respect to R and h λ ( x ) ∈ R for each x ∈ g ( P ) and λ ∈ Λ ∩ R . Clearly, a lift of a real holomorphic motion is again real. Example 4.1.
Let g be a marked map with P ⊃ P = { c = 0 } , so that P = { c , . . . , c q − } , c i = g i ( c ) , ≤ i < q are distinct and g q ( c ) = c . Consider a deformation ( g, G, p ) W where W is a neighbourhood of c and let p : W → C be so that p ≡ . Consider the holomorphicmotion of g ( P ) = P defined by h λ ( c i ) = c i + v i λ . Then ˆ h λ ( c i ) = c i + ˆ v i λ + O ( λ ) is defined by ˆ h λ ( c ) = 0 , G h λ ( c )) (ˆ h λ ( c i )) = h λ ( c i +1 ) = c i +1 + v i +1 λ , i = 1 , . . . , q − where we take c q = c , v q = v . Writing L i = ∂G w ( c i ) ∂w and D i = Dg ( c i ) we obtain ˆ v = 0 , L i v + D i ˆ v i = v i +1 . So A = . . . − L /D /D − L /D /D . . . ... ... ... ... . . . ... − L q − /D q − . . . /D q − /D q − − L q − /D q − . . . . Hence I − ρ A = . . .
00 1 + ρL /D − ρ/D ρL /D − ρ/D . . . ... ... ... ... . . . ... ρL q − /D q − . . . − ρ/D q − − ρ/D q − ρL q − /D q − . . . . So (4.1) det( I − ρ A ) = 1 + L D ρ + L D D ρ + · · · + L q − D D . . . D q − ρ q − . So if the spectrum of A is contained in the open unit disc and L i , D i are real, then (4.1) is strictlypositive for all ρ ∈ [ − , . Note that when G w ( z ) = g ( z ) + ( w − c ) , the expression (4.1) agreeswith (2.4) for ρ = 1 . Relating the transfer operator with transversality.
It turns out that transversalityis closely related to the eigenvalues of A : Theorem 4.1.
Assume the following holds: for any r < j ≤ ν , Dg q j − l j ( c l j ,j ) (cid:54) = 1 . Then thefollowing statements are equivalent:(1) is an eigenvalue of A ;(2) D R ( c ) is degenerate.More precisely, the dimension of kernel of D R ( c ) is equal to the dimension of the eigenspace of A associated with eigenvalue .Proof. We first show that (1) implies (2), even without the assumption. So suppose that is aneigenvalue of A and let v = ( v ( x )) x ∈ g ( P ) be an eigenvector associated with . For t ∈ D , define h t ( x ) = x + tv ( x ) for each x ∈ g ( P ) and w ( t ) = ( c ,j + tv ( c ,j )) νj =1 . Then for each x ∈ g ( P ) \ P ,(4.2) G w ( t ) ( h t ( x )) − h t ( g ( x )) = O ( t ) , and for each x = c ,j ∈ g ( P ) ∩ P , we have(4.3) h t ( x ) − p j ( w ( t )) = O ( t ) . For each ≤ j ≤ ν , and each ≤ k < q j , applying (4.2) repeatedly, we obtain(4.4) G kw ( t ) ( h t ( c ,j )) = h t ( g k ( c ,j )) + O ( t ) . Together with (4.3), this implies that R j ( w ( t )) = O ( t ) , holds for all ≤ j ≤ ν . It remains to show w (cid:48) (0) (cid:54) = . Indeed, otherwise, by (4.4), it wouldfollow that v ( g k ( c ,j )) = ( g k ) (cid:48) ( c ,j ) v ( c ,j ) = 0 for each ≤ j ≤ ν and ≤ k < q j , and hence v ( x ) = 0 for all x ∈ g ( P ) , which is absurd. We completed the proof that (1) implies (2).Now let us prove that (2) implies (1) under the assumption of the lemma. Suppose that D R ( c ) is degenerate. Then there exists a non-zero vector ( w , w , · · · , w ν ) in C ν such that R j ( w ( t )) = O ( t ) as t → for all j = 1 , . . . , ν , where w ( t ) = ( w j ( t )) νj =1 = ( c ,j + tw j ) νj =1 . Weclaim that w j = w j (cid:48) holds whenever c ,j = c ,j (cid:48) , ≤ j, j (cid:48) ≤ ν . Indeed, Case 1. If ≤ j ≤ r then ≤ j (cid:48) ≤ r and µ ( j ) = µ ( j (cid:48) ) , q j = q j (cid:48) . Then G q j − w ( t ) ( w j ( t )) − G q j − w ( t ) ( w j (cid:48) ( t )) = R j ( w ( t )) − R j (cid:48) ( w ( t )) = O ( t ) which implies that w j ( t ) − w j (cid:48) ( t ) = O ( t ) , i.e. w j = w j (cid:48) . Case 2. If r < j ≤ ν then r < j (cid:48) ≤ ν and ˆ l j = ˆ l j (cid:48) , ˆ q j = ˆ q j (cid:48) where we define for any r < j ≤ ν the integers ˆ l j < ˆ q j minimal so that g ˆ q j ( c ,j ) = g ˆ l j ( c ,j ) . By the chain rule it follows that G q j − w ( t ) ( w j ( t )) − G l j − w ( t ) ( w j ( t )) = O ( t ) implies G ˆ q j − w ( t ) ( w j ( t )) − G ˆ l j − w ( t ) ( w j ( t )) = O ( t ) under the assumption Dg q j − l j ( c l j ,j ) (cid:54) = 1 .Thus we obtain G ˆ q j − w ( t ) ( w j ( t )) − G ˆ q j − w ( t ) ( w j (cid:48) ( t )) = G ˆ l j − w ( t ) ( w j ( t )) − G ˆ l j − w ( t ) ( w j (cid:48) ( t )) + O ( t ) , which implies that ( Dg ˆ q j − ( c l j ,j ) − Dg ˆ l j − ( c l j ,j ))( w j ( t ) − w j (cid:48) ( t )) = O ( t ) . If such j and j (cid:48) exist then c l j ,j is a hyperbolic periodic point, hence Dg ˆ q j − ( c l j ,j ) (cid:54) = Dg ˆ l j − ( c l j ,j ) .It follows that w j = w j (cid:48) .Thus the Claim is proved. To obtain an eigenvector for A with eigenvalue , define v ( c ,j ) = w j , v ( c ,j ) = ddt p j ( w ( t )) | t =0 . For points x ∈ g ( P ) \ P , there is j and ≤ s < q j such that x = g s ( c ,j ) , define v ( x ) = ddt G s − w ( t ) ( w j ( t )) | t =0 . Note that v ( x ) does not depend on the choice of j and s . (This can be proved similarly as the claim.)The above argument builds an isomorphism between { v ∈ C ν : D R ( c , v ) = 0 } and theeigenspace of A associated with eigenvalue . So these two spaces have the same dimension. (cid:3) The spectrum of A and the determinant of some matrix D ( ρ ) . Define D ( ρ ) =( D j,k ( ρ )) ≤ j,k ≤ ν as follows: Put L k ( z ) = ∂G w ( z ) ∂w k | w = c ; p j,k = ∂p j ∂w k ( c ); L j,k = 0 and L mj,k = m (cid:88) n =1 ρ n L k ( c n,j ) Dg n ( c ,j ) for m > D jk ( ρ ) = δ jk + L q j − j,k − ρ q j p µ ( j ) ,k Dg q j − ( c ,j ) when ≤ j ≤ r and D jk ( ρ ) = δ jk + L q j − j,k − ρ q j − l j Dg q j − l j ( c l j ,j ) (cid:16) L l j − jk + δ j,k (cid:17) when r < j ≤ ν . Note that(4.5) det( D R ( c )) = ν (cid:89) j =1 Dg q j − ( c ,j ) det( D (1)) . We say that ρ ∈ C is an exceptional value if there exists r < j ≤ ν such that Dg q j − l j ( c l j ,j ) = ρ q j − l j . Proposition 4.2.
For each non-exceptional ρ ∈ C , we have (4.6) det( I − ρ A ) = 0 ⇔ det( D ( ρ )) = 0 . Proof.
For ρ = 0 , det( I ) = det( D (0)) = 1 . Assume ρ (cid:54) = 0 . Define a new triple ( g ρ , G ρ , p ρ ) asfollows. • For each x ∈ P \ P , G ρw ( z ) = G w ( x ) + Dg ( x ) ρ ( z − x ) in a neighbourhood of x ; • g ρ ( x ) = g ( x ) for each x ∈ P and g ρ ( z ) = G ρ c ( z ) in a neighbourhood of P \ P ; • p ρ ( w ) = c + ρ ∂ p ∂w ( c ) · ( w − c ) . Let A ρ be the transfer operator associated with the triple ( g ρ , G ρ , p ρ ) . Then it is straightforwardto check that A ρ = ρ A . We can define a map R ρ = ( R ρ , R ρ , · · · , R ρν ) for each ρ (cid:54) = 0 in the obvious way: R ρj ( w ) = ( G ρw ) q j − ( w j ) − p ρµ ( j ) ( w ) for ≤ j ≤ r and R ρj ( w ) = ( G ρw ) q j − ( w j ) − ( G ρw ) l j − ( w j ) for r < j ≤ ν . As long as ρ is non-exceptional for the triple ( g, G, p ) , the new triple ( g ρ , G ρ , p ρ ) satisfies the assumption of Theorem 4.1, thus det( I − A ρ ) = 0 ⇔ D R ρ ( c ) is degenerate . Direct computation shows that the ( j, k ) -th entry of D R ρ ( c ) is equal to D j,k ( ρ ) Dg q j − ( c ,j ) /ρ q j − . Indeed, for each ≤ j ≤ r , D ρj,k ( ρ ) = ∂ ( G ρw ) q j − ( c ,j ) ∂w k | w = c + Dg q j − ( c ,j ) ρ q j − δ jk − ρ ∂p µ ( j ) ∂w k = Dg q j − ( c ,j ) ρ q j − δ jk + q j − (cid:88) n =1 ρ n L k ( c n,j ) Dg n ( c ,j ) − ρ q j p µ ( j ) ,k Dg q j − ( c ,j ) , and for r < j ≤ ν , D ρjk ( ρ )= ∂ (( G ρw ) q j − ( c ,j ) − ( G ρw ) l j − ( c ,j )) ∂w k | w = c + δ jk (cid:18) Dg q j − ( c ,j ) ρ q j − − Dg l j − ( c ,j ) ρ l j − (cid:19) = Dg q j − ( c ,j ) ρ q j − L q j − j,k − Dg l j − ( c ,j ) ρ l j − L l j − j,k + δ jk (cid:18) Dg q j − ( c ,j ) ρ q j − − Dg l j − ( c ,j ) ρ l j − (cid:19) Therefore det( I − ρ A ) = 0 if and only if det( D ( ρ )) = 0 . (cid:3) Positive transversality in the real case.
To illustrate the power of the previous propo-sition we state:
Corollary 4.3 (Positive transversality) . Let ( g, G, p ) W be a real local holomorphic deformationof a real marked map g . Assume that one has | Dg q j − l j ( c l j ,j ) | > for all r < j ≤ ν . Assumefurthermore that all the eigenvalues of A lie in the set {| ρ | ≤ , ρ (cid:54) = 1 } . Then the ‘positivelyoriented’ transversality condition holds. Proof.
Write the polynomial det( D ( ρ )) in the form (cid:81) Ni =1 (1 − ρρ i ) , where ρ i ∈ C \ { } . Because of(4.5) it suffices to show that det( D (1)) > . Since det( D ( ρ )) is a real polynomial in ρ , this followsfrom ρ i (cid:54)∈ [1 , ∞ ) for each i . Arguing by contradiction, assume that ρ i ≥ for some i . Then /ρ i is a zero of det( D ( ρ )) . As | /ρ i | ≤ , /ρ i is not an exceptional value. Thus det( I − A /ρ i ) = 0 ,which implies that ρ i ≥ is an eigenvalue of A , a contradiction! (cid:3) A remark on an alternative transfer operator.
Proposition 4.2 shows that for non-exceptional ρ , one has (4.6). One can also associate to ( g, G, p ) another linear operator A J forwhich(4.7) det D ( ρ ) = det( I − ρ A J ) holds for all ρ ∈ C . Here J denotes a collection of all pairs ( i, j ) such that ≤ j ≤ ν , ≤ i ≤ q j − and if i = 0 then j = µ ( j (cid:48) ) for some ≤ j (cid:48) ≤ ν . Given a collection of functions { c i,j ( λ ) } ( i,j ) ∈ J which are holomorphic in a small neighbourhood of λ = 0 , there is anothercollection of holomorphic near functions { ˆ c i,j ( λ ) } ( i,j ) ∈ J such that ˆ c ,j ( λ ) = p j ( c ( λ )) where c ( λ ) = ( c , ( λ ) , · · · , c ,ν ( λ )) and, for i (cid:54) = 0 , G ( c ( λ ) , ˆ c i,j ) = c i +1 ,j ( λ ) . Here we set c q j ,j ( λ ) = c ,µ ( j ) ( λ ) for ≤ j ≤ r and c q j ,j ( λ ) = c l j ,j ( λ ) for r < j ≤ ν . Define the linear map A J : C J → C J by taking the derivative at λ = 0 : A J ( { c (cid:48) i,j (0) } ( i,j ) ∈ J ) = { ˆ c (cid:48) i,j (0) } ( i,j ) ∈ J . Explicitly, we get: ˆ c (cid:48) i,j (0) = (cid:80) νk =1 p j,k if i = 0 and j = µ ( j (cid:48) ) for some j (cid:48) Dg ( c i,j ) ( v i +1 ,j − (cid:80) νk =1 L k ( c i,j ) v ,k ) if ≤ i < q j − , ≤ j ≤ ν Dg ( c qj − ,j ) (cid:0) v ,µ ( j ) − (cid:80) νk =1 L k ( c q j − ,j ) v ,k (cid:1) if i = q j − , ≤ j ≤ r Dg ( c qj − ,j ) (cid:0) v l j ,j − (cid:80) νk =1 L k ( c q j − ,j ) v ,k (cid:1) if i = q j − , r < j ≤ ν Elementary properties of determinants being applied to the matrix I − ρ A J lead to (4.7). Observethat A J = A if (and only if) all points c i,j , ( i, j ) ∈ J are pairwise different. Therefore, we have: det( I − ρ A ) = det D ( ρ ) for every ρ ∈ C provided (cid:80) νj =1 ( q j −
1) + r = P .5. The lifting property and the spectrum of A The next proposition shows that the lifting property implies that the spectrum of A is in theclosed unit disc. Proposition 5.1. If ( g, G, p ) W has the lifting property, then the spectral radius of the associatedtransfer operator A is at most and every eigenvalue of A of modulus one is semisimple (i.e.its algebraic multiplicity coincides with its geometric multiplicity). Moreover, for ( g, G, p ) W real,we only need to assume that the lifting property with respect to real holomorphic motions.Proof. For any v = ( v ( x )) x ∈ g ( P ) , construct a holomorphic motion h (0) λ over (Λ , for somedomain Λ (cid:51) , such that ddλ h (0) λ ( x ) | λ =0 = v ( x ) for all x ∈ g ( P ) . Then A k ( v ) = (cid:18) ddλ h ( k ) λ ( x ) | λ =0 (cid:19) x ∈ g ( P ) for every k > . By Cauchy’s integral formula, there exists C = C ( M, ε ) such that | ddλ h ( k ) λ ( x ) | λ =0 | ≤ C holds for all x ∈ g ( P ) and all k . It follows that for any v ∈ C g ( P ) , A k ( v ) is a bounded se-quence. Thus the spectral radius of A is at most one and every eigenvalue of A of modulus oneis semisimple.Suppose ( g, G, p ) W is real. Then for any v ∈ R g ( P ) , the holomorphic motion h (0) λ can bechosen to be real. Thus if ( g, G, p ) W has the real lifting property, then {A k ( v ) } ∞ k =0 is boundedfor each v ∈ R g ( P ) . The conclusion follows. (cid:3) To obtain that the radius is strictly smaller than one, we shall apply the argument to a suitableperturbation of the map g . For example, we have the following: Proposition 5.2 (Robust spectral property) . Let ( g, G, p ) W be as above. Let Q be a polynomialsuch that Q ( c ,j ) = 0 for ≤ j ≤ ν and Q ( x ) = 0 , Q (cid:48) ( x ) = 1 for every x ∈ g ( P ) . Let ϕ ξ ( z ) = z − ξQ ( z ) and for ξ ∈ (0 , let ψ ξ ( w ) = ( ϕ − ξ ( w ) , · · · , ϕ − ξ ( w ν )) be a map from aneighbourhood of c into a neighbourhood of c . Suppose that there exists ξ ∈ (0 , such that thetriple ( ϕ ξ ◦ g, ϕ ξ ◦ G, p ◦ ψ ξ ) has the lifting property. Then the spectral radius of A is at most − ξ .Proof. Note that ˜ g := ϕ ξ ◦ g is a marked map with the same sets P ⊂ P . Furthermore, ˜ g i ( c ,j ) = g i ( c ,j ) = c i,j , D (cid:101) g ( c i,j ) = (1 − ξ ) Dg ( c i,j ) , ∂ϕ ξ ◦ G∂w k ( c , z ) = (1 − ξ ) ∂G∂w k ( c , z ) for each z ∈ g ( P ) \ P , and p ◦ ψ ξ ∂w k ( c ) = (1 − ξ ) − ∂ p ∂w k ( c ) . Therefore, the operator which is associatedto the triple ( ϕ ξ ◦ g, ϕ ξ ◦ G, p ◦ ψ ξ ) is equal to (1 − ξ ) − A , Since the latter triple has the liftingproperty, by Proposition 5.1, the spectral radius of (1 − ξ ) − A is at most . (cid:3) For completeness we include:
Lemma 5.3.
Assume that the spectrum radius of A is strictly less than . Then the liftingproperty holds.Proof. Let Φ( Z ) = ( ϕ x ( Z )) x ∈ g ( P ) be the holomorphic map defined from a neighbourhood V ofthe point z := g ( P ) ∈ C g ( P ) by G Z ( ϕ x ( Z )) = z g ( x ) , x ∈ g ( P ) \ P ,ϕ c ,j ( Z ) = p j ( Z ) , ≤ j ≤ ν, where z = ( z c ,j ) νj =1 , Z = ( z x ) x ∈ g ( P ) . Then Φ( z ) = z . Moreover, if h λ is a holomorphic motionof g ( P ) over (Λ , with h λ = ( h λ ( x )) x ∈ g ( P ) ∈ V for all λ , then (cid:98) h λ ( x ) := (Φ( h λ )) x is the lift of h λ over (Λ , .So the derivative of Φ at z is equal to A , and hence z is a hyperbolic attracting fixed pointof Φ . Therefore, there exist N > and a neighborhood U of z such that Φ N is well-defined on U and such that Φ N ( U ) is compactly contained in U . It follow Φ n converges uniformly to theconstant z in U .Let us prove that ( g, G, p ) W has the lifting property. Indeed, if h λ is a holomorphic motionof g ( P ) over ( D , , then there exists ε > such that h λ := ( h λ ( x )) x ∈ g ( P ) ∈ U , so that h ( k ) λ :=Φ k ( h λ ) is well-defined. Let h ( k ) λ ( x ) , x ∈ g ( P ) , be such that h ( k ) λ = ( h ( k ) λ ( x )) x ∈ g ( P ) . Then for each k ≥ , h ( k ) λ is a holomorphic motion of g ( P ) over ( D ε , and h ( k +1) λ is the lift of h ( k ) λ . (cid:3) The lifting property and persistence of critical relations
The main technical result in this paper is the following theorem:
Theorem 6.1.
Assume that either the triple ( g, G, p ) W has the lifting property or ( g, G, p ) W isreal and has the real lifting property. Assume also that for all r < j ≤ ν , Dg q j − l j ( c l j ,j ) (cid:54) = 1 .Then(1) All eigenvalues of A are contained in D .(2) There is a neighborhood W (cid:48) of c in W such that (6.1) { w ∈ W (cid:48) | R ( w ) = 0 } is a smooth submanifold of W (cid:48) , and its dimension is equal to the geometric multiplicityof the eigenvalue of A . The second statement is useful to conclude that D R is non-degenerate at c , or equivalently,that is not an eigenvalue of A . Indeed, if ν = 1 and if is an eigenvalue of A , the manifold (6.1)must contain a neighbourhood of c and hence R ( w ) = 0 holds for every w ∈ C near c ∈ C ,which only happens for trivial family ( g, G, p ) W . It is also possible to apply this statement ina more subtle way, see [24].Let Λ be a domain in C which contains . A holomorphic motion h λ ( x ) of g ( P ) over (Λ , iscalled asymptotically invariant of order m (with respect to ( g, G, p ) W ) if there is a subdomain Λ ⊂ Λ which contains and a holomorphic motion (cid:98) h λ ( x ) which is the lift of h λ over (Λ , ,such that(6.2) (cid:98) h λ ( x ) − h λ ( x ) = O ( λ m +1 ) as λ → . Obviously,
Lemma 6.1. is an eigenvalue of A if and only if there is a non-degenerate holomorphic motionwhich is invariant of order . Here, a holomorphic motion h λ ( x ) is called non-degenerate if ddλ h λ ( x ) | λ =0 (cid:54) = 0 holds for some x ∈ g ( P ) .A crucial step in proving this theorem is the following Lemma 6.4 whose proof requires thefollowing easy fact and its corollary. Fact 6.2.
Let F : U → C be a holomorphic function defined in an open set U of C N , N ≥ .Let γ, (cid:101) γ : D ε → U be two holomorphic curves such that γ ( λ ) − (cid:101) γ ( λ ) = O ( λ m +1 ) as λ → . Then F ( γ ( λ )) − F ( (cid:101) γ ( λ )) = N (cid:88) i =1 ∂F∂z i ( γ (0))( γ i ( λ ) − (cid:101) γ i ( λ )) + O ( λ m +2 ) as λ → . Proof.
For fixed λ small, define δ ( t ) = (1 − t ) (cid:101) γ ( λ ) + tγ ( λ ) and f ( t ) = F ( δ ( t )) . Then f (cid:48) ( t ) = N (cid:88) i =1 ∂F∂z i ( δ ( t ))( γ i ( λ ) − (cid:101) γ i ( λ )) . Since δ ( t ) − γ (0) = O ( λ ) , and F ( γ ( λ )) − F ( (cid:101) γ ( λ )) = (cid:82) f (cid:48) ( t ) dt , the equality follows. (cid:3) Corollary 6.3.
A holomorphic motion h λ of g ( P ) is asymptotically invariant of order m if andonly if(1) For each x ∈ g ( P ) \ P , G h λ ( c , ) , ··· ,h λ ( c ,ν ) ( h λ ( x )) = h λ ( g ( x )) + O ( λ m +1 ) as λ → . (2) For x = c ,j ∈ g ( P ) , p j ( h λ ( c , ) , · · · , h λ ( c ,ν )) = h λ ( c ,j ) + O ( λ m +1 ) as λ → . Proof. If h λ is asymptotically invariant of order m , we get (1) applying Fact 6.2 to the func-tion F ( z , z , · · · , z ν , z ν +1 ) = G ( z ,z , ··· ,z ν ) ( z ν +1 ) , and we get (2) applying it to the function F ( z , z , · · · , z ν ) = p j ( z , z , · · · , z ν ) . Vice versa, assume that (1)-(2) hold. Given x ∈ g ( P ) \ P ,let F ( z , z , · · · , z ν , z ν +1 ) be a local branch of G − z ,z , ··· ,z ν ) ( z ν +1 ) which is a well defined holo-morphic function in a neighborhood of ( c , g ( x )) . Let V ( λ ) = G h λ ( c , ) , ··· ,h λ ( c ,ν ) ( h λ ( x )) and ˆ V ( λ ) = h λ ( g ( x )) . By (1), V ( λ ) − ˆ V ( λ ) = O ( λ m +1 ) . Hence, by Fact 6.2, h λ ( x ) − ˆ h λ ( x ) = F ( h λ ( c , ) , · · · , h λ ( c ,ν ) , V ( λ )) − F ( h λ ( c , ) , · · · , h λ ( c ,ν ) , ˆ V ( λ )) = O ( λ m +1 ) . For x = c ,j , the claim is straightforward. (cid:3) Lemma 6.4.
One has the following:(1) Assume ( g, G, p ) W has the lifting property. Suppose that there is a holomorphic motion h λ of g ( P ) over (Λ , which is asymptotically invariant of order m for some m ≥ .Then there is a non-degenerate holomorphic motion H λ of g ( P ) over some ( ˜Λ , whichis asymptotically invariant of order m + 1 . Besides, H λ ( x ) − h λ ( x ) = O ( λ m +1 ) as λ → for all x ∈ g ( P ) .(2) Assume ( g, G, p ) W is real and has the real lifting property. Suppose that there is a realholomorphic motion h λ of g ( P ) over (Λ , which is asymptotically invariant of order m for some m ≥ . Then there is a non-degenerate real holomorphic motion H λ of g ( P ) oversome ( ˜Λ , which is asymptotically invariant of order m + 1 . Besides, H λ ( x ) − h λ ( x ) = O ( λ m +1 ) as λ → for all x ∈ g ( P ) . Proof.
We shall only prove the first statement as the proof of the second is the same with obviouschange of terminology. Let h λ be a non-degenerate holomorphic motion of g ( P ) over (Λ , whichis asymptotically invariant of order m . By assumption that ( g, G, p ) W has the lifting property ,there exists a smaller domain Λ ⊂ Λ and holomorphic motions h ( k ) λ over Λ , k = 0 , , . . . suchthat h (0) λ = h λ and such that h ( k +1) λ is the lift of h ( k ) λ over (Λ , for each k ≥ . Moreover, thefunctions h ( k ) λ are uniformly bounded. For each k ≥ , define ψ ( k ) λ ( x ) = 1 k k − (cid:88) i =0 h ( i ) λ ( x ) , and ϕ ( k ) λ ( x ) = 1 k k (cid:88) i =1 h ( i ) λ ( x ) . By shrinking Λ , we may assume that there exists k n → ∞ , such that ψ ( k n ) λ ( x ) converges uni-formly in λ ∈ Λ as k n → ∞ to a holomorphic function H λ ( x ) . Shrinking Λ furthermore ifnecessary, H λ defines a holomorphic motion of g ( P ) over (Λ , . Clearly, ϕ ( k n ) λ ( x ) convergesuniformly to H λ ( x ) as well.Let us show that H λ is asymptotically invariant of order m + 1 by applying the fact above.Due to Corollary 6.3 (and taking k = k n → ∞ in the next equations) this amounts to showing:(i) For each x ∈ g ( P ) \ P , and any k ≥ , G ψ ( k ) λ ( c , ) , ··· ,ψ ( k ) λ ( c ,ν ) ( ϕ ( k ) λ ( x )) = ψ ( k ) λ ( g ( x )) + O ( λ m +2 ) as λ → . (ii) For x = c ,j ∈ g ( P ) , p j ( ψ ( k ) λ ( c , ) , · · · , ψ ( k ) λ ( c ,ν )) = ϕ ( k ) λ ( c ,j ) + O ( λ m +2 ) as λ → . Let us prove (i). Fix x ∈ g ( P ) \ P and k ≥ . Let F ( z , z , · · · , z ν , z ν +1 ) = G ( z ,z , ··· ,z ν ) ( z ν +1 ) .By the construction of h ( k ) λ , we have F ( h ( i ) λ ( c , ) , · · · , h ( i ) λ ( c ,ν ) , h ( i +1) λ ( x )) = h ( i ) λ ( g ( x )) for every i ≥ . Thus(6.3) ψ ( k ) λ ( g ( x )) = 1 k k − (cid:88) i =0 F ( h ( i ) λ ( c , ) , · · · , h ( i ) λ ( c ,ν ) , h ( i +1) λ ( x )) . Since all the functions h ( i ) λ ( x ) , ψ ( k ) λ ( x ) , ϕ ( k ) λ ( x ) have the same derivatives up to order m at λ = 0 ,applying Fact 6.2, we obtain F ( h ( i ) λ ( c , ) , · · · , h ( i ) λ ( c ,ν ) , h ( i +1) λ ( x )) − F ( ψ ( k ) λ ( c , ) , · · · , ψ ( k ) λ ( c ,ν ) , ϕ ( k ) λ ( x ))= ν (cid:88) j =1 ∂F∂z j ( c , x )( h ( i ) λ ( c ,j ) − ψ ( k ) λ ( c ,j )) + ∂F∂z ν +1 ( c , x )( h ( i +1) λ ( x ) − ϕ ( k ) λ ( x )) + O ( λ m +2 ) , as λ → . Summing over i = 0 , , · · · , k − and using the definition of ψ ( k ) λ ( x ) and ϕ ( k ) λ ( x ) weobtain k k − (cid:88) i =0 F ( h ( i ) λ ( c , ) , · · · , h ( i ) λ ( c ,ν ) , h ( i +1) λ ( x ))= F ( ψ ( k ) λ ( c , ) , · · · , ψ ( k ) ( c ,ν ) , ϕ ( k ) λ ( x )) + O ( λ m +2 ) . Together with (6.3), this implies the equality in (i).For (ii), we use F ( z , · · · , z ν ) = p j ( z , · · · , z ν ) and argue in a similar way. (cid:3) Proof of the Main Theorem.
By Proposition 5.1, all eigenvalues of A are contained in D . Itremains to prove (2). Let L = { v ∈ C ν : D R ( c , v ) = 0 } and let d be the dimension of L .By Theorem 4.1, L has the same dimension as the eigenspace of A associated with eigenvalue . Moreover, by Lemma 6.4, for each m ≥ and each v ∈ L , there is a holomorphic motion h λ ( x ) of g ( P ) over D ε for some ε > which is asymptotic invariant of order m and satisfies h λ ( c ,j ) = c ,j + v ( c ,j ) λ + O ( λ ) as λ → . Putting w j ( λ ) = h λ ( c ,j ) , we obtain a holomorphiccurve λ (cid:55)→ w ( λ ) = ( w j ( λ )) νj =1 , λ ∈ D ε , such that w (cid:48) j (0) = v ( c ,j ) and R j ( w ( λ )) = O ( λ m +1 ) for all j = 1 , . . . , ν. If d = 0 , i.e., L = { } , then D R ( c ) is invertible, so R is a local diffeomorphism, and for asmall neighborhood W (cid:48) of c , the set in (6.1) consists of a single point c .Now assume d = ν , i.e., L = C ν . We claim that R ( w ) ≡ . Otherwise, there exists m ≥ suchthat R ( w ) = (cid:80) ∞ k = m P k ( w − c ) in a neighborhood of c , where P k ( u ) is a homogeneous polynomialin u of degree k and P m ( u ) (cid:54)≡ . Therefore, there exists v ∈ C d such that P m ( λ v ) = Aλ m forsome A (cid:54) = 0 . By the argument above, there is holomorphic curve λ (cid:55)→ w ( λ ) passing through c and tangent to v at λ = 0 , such that |R ( w ( λ )) | = O ( λ m +1 ) . However, R ( w ( λ )) = P m ( w ( λ ) − c )) + (cid:88) k>m P k ( w − c ( λ )) = Aλ m + O ( λ m +1 ) , a contradiction.The case < d < ν can be done similarly. To be definite, let us assume that(6.4) ∂ ( R , R , · · · , R d (cid:48) ) ∂ ( w , w , · · · , w d (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) w = c (cid:54) = 0 , where d (cid:48) = ν − d . By the Implicit Function Theorem, there is holomorphic map Φ : B → C d (cid:48) ,where B is a neighborhood of u = ( c ,d (cid:48) +1 , c ,d (cid:48) +2 , · · · , c ,ν ) in C d such that(6.5) R j (Φ( u ) , u ) = 0 for all ≤ j ≤ d (cid:48) , u ∈ B, and t = Φ( u ) is the only solution of R j ( t, u ) = 0 , ≤ j ≤ d (cid:48) , in a fixed neighborhood of ( c , , c , , · · · , c ,d ) . It suffices to prove that R j (Φ( u ) , u ) = 0 for u close to u , d (cid:48) < j ≤ ν. To this end, we only need to show that for any m ≥ , and any e ∈ C d , there is a holomorphiccurve u ( λ ) in B which passes through u at λ = 0 , such that u (cid:48) (0) = e and(6.6) R j (Φ( u ( λ )) , u ( λ )) = O ( λ m +1 ) , as λ → , d (cid:48) < j ≤ ν. Indeed, the curve ˜ w ( λ ) = (Φ( u + λ e ) , u + λ e ) is tangent to L at c . Thus by the argumentin the first paragraph of the proof, there is a curve w ( λ ) = ( t ( λ ) , u ( λ )) ∈ C d (cid:48) × C d , tangent to ˜ w ( λ ) at λ = 0 such that(6.7) R j ( w ( λ )) = O ( λ m +1 ) as λ → , ≤ j ≤ ν. Together with (6.4) and (6.5), this implies that | Φ( u ( λ )) − t ( λ ) | = O ( λ m +1 ) . Finally, by (6.7), we obtain (6.6), completing the proof of the Main Theorem. (cid:3) Families of the form f λ ( x ) = f ( x ) + λ and f λ ( x ) = λf ( x ) In this section we will apply these techniques to show that one has monotonicity and thetransversality properties (2.4) and (2.3) within certain families of real maps of the form f λ ( x ) = f ( x ) + λ and f λ ( x ) = λ · f ( x ) where x (cid:55)→ f ( x ) has one critical value (and is unimodal - possiblyon a subset R ) or satisfy symmetries. There are quite a few papers giving examples for whichone has non-monotonicity for such families, see for example [4, 21, 39, 50]. In this section wewill prove several theorems which show monotonicity for a fairly wide class of such families.In Subsection 7.1 we show that the methods we developed in the previous section apply ifone has something like a polynomial-like map f : U → V with sufficiently ‘big complex bounds’. This gives yet another proof for monotonicity for real families of the form z (cid:96) + c , c ∈ R in thesetting when (cid:96) is an even integer. We also apply this method to a family of maps with a flatcritical point in Subsection 7.2. In Subsection 7.3 we show how to obtain the lifting property inthe setting of one parameter families of the form f a ( x ) = af ( x ) with f in some rather generalclass of maps.7.1. Families of the form f λ ( x ) = f ( x ) + λ with a single critical point. Let f : U → V bea map from the class F defined in Subsection 2.8. Consider a marked map g with g = f + g (0) for some f ∈ F from a finite set P into itself with P ⊃ P = { } , P \ P ⊂ U . In other words, g extends to a holomorphic map g : U g → V g where • U g is a bounded open set in C such that U g ⊃ P \ { } and ∈ U g ; • V g is a bounded open set in C such that c := g (0) ∈ V g ; • g : U g \ { } → V g \ { c } is an unbranched covering; • V g ⊃ B ( c ; diam( U g )) ⊃ U g .Next define a local holomorphic deformation ( g, G, p ) W of g as follows: G w ( z ) = g ( z )+( w − g (0)) and p ( w ) = 0 for all w ∈ W := C . Theorem 7.1.
Let ( g, G, p ) W be as above. Then(1) ( g, G, p ) W satisfies the lifting property;(2) the spectrum of the operator A is contained in D \ { } .If, in addition, the robust separation property V g ⊃ B ( c ; diam( U g )) ⊃ U g holds, then the spectralradius of A is strictly smaller than and det( I − ρ A ) = q − (cid:88) i =0 ρ i Dg i ( c ) (cid:54) = 0 holds for all | ρ | ≤ . In particular, if g, G are real then (cid:80) q − i =0 1 Dg i ( c ) > .Proof. Let us show that ( g, G, p ) C satisfies the lifting property. For each domain ∆ (cid:51) in C ,let M ∆ denote the collection of all holomorphic motions h λ of g ( P ) over (∆ , such that for all λ ∈ ∆ ,(7.1) h λ ( x ) ∈ U g for all x ∈ g ( P ) \ { } and h λ (0) = 0 . Claim.
Let ∆ (cid:51) be a simply connected domain in C . Then any holomorphic motion h λ in M ∆ has a lift (cid:98) h λ which is again in the class M ∆ .Indeed, g ( P \ { } ) ⊂ g ( U g \ { } ) = V g \ { g (0) } . So for each x ∈ P \ { } , g ( x ) (cid:54) = g (0) , thus forany λ ∈ ∆ , < | h λ ( g ( x )) − h λ ( g (0)) | < diam( U g ) , hence by V g ⊃ B ( c ; diam( U g )) ⊃ U g , h λ ( g ( x )) − h λ ( g (0)) + g (0) ∈ V g \ { g (0) } . Since g : U g \ { } → V g \ { g (0) } is an unbranched covering and ∆ is simply connected, there isa holomorphic function λ (cid:55)→ (cid:98) h λ ( x ) , from ∆ to U g \ { } , such that (cid:98) h ( x ) = x and g ( (cid:98) h λ ( x )) = h λ ( g ( x )) − h λ ( g (0)) + g (0) , i.e. G h λ ( g (0)) ( (cid:98) h λ ( x )) = h λ ( g ( x )) . Define (cid:98) h λ (0) = 0 if ∈ g ( P ) . Then (cid:98) h λ is a lift of h λ over ∆ .For any holomorphic motion h λ of g ( P ) over (Λ , with h λ (0) = 0 , there is a simply connectedsub-domain ∆ (cid:51) such that the restriction of h λ to ∆ belongs to the class M ∆ . It follows that ( g, G, p ) W has the lifting property.Therefore the assumptions of the Main Theorem are satisfied. The operator A cannot have aneigenvalue because otherwise for all parameters w ∈ W the G w would have the same dynamics.Hence, (2) in the conclusion of the theorem follows.If the robust separation property V g ⊃ B ( c ; diam( U g )) ⊃ U g holds, then Proposition 5.2applies and therefore the spectral radius of A is strictly smaller than . As in Example 4.1 theconclusion follows. (cid:3) Corollary 7.1.
For any even integer d , transversality condition (2.3) holds and the topologicalentropy of g c ( z ) = z d + c depends monotonically on c ∈ R . A unimodal family map f ∈ F with a flat critical point. Fix (cid:96) ≥ , b > e(cid:96) ) /(cid:96) andconsider f c ( x ) = (cid:26) be − / | x | (cid:96) + c for x ∈ R \ { } ,c for x = 0 . Note that R + (cid:51) x (cid:55)→ xe /x (cid:96) has a unique critical point at x = (cid:96) /(cid:96) corresponding to a minimumvalue (cid:96)e ) /(cid:96) . Therefore the assumption on b implies that b = 2 xe /x (cid:96) has a unique solution x = β ∈ (0 , (cid:96) /(cid:96) ) . This implies in particular that the map f − β has the Chebeshev combinatorics: f − β (0) = − β and f − β ( β ) = β . Note that Df − β ( β ) = be − /β (cid:96) (cid:96)β (cid:96) +1 = 2 (cid:96)β (cid:96) > . Therefore, there exists x > x > β such that f − β ( x ) = x and x − β > x − β ) . Choosing x close enough to β , we have R := f ( x ) = x + β < b. For a bounded open interval J ⊂ R , let D ∗ ( J ) denote the Euclidean disk with J as a diameter.This set corresponds to the set of points for which the distance to J w.r.t. the Poincaré metricon C J = C \ ( R \ J ) is at most some k > . Also, let B ∗ ( x, R ) = B ( x, R ) \ { x } where B ( x, R ) is the open disc with radius R and centre at x . Lemma 7.2.
The map f : ( − x , ∪ (0 , x ) → (0 , R ) extends to an unbranched holomor-phic covering map F : U → B ∗ (0 , R ) , where U ⊂ D ∗ (( − x , ∪ D ∗ ((0 , x )) . In particular, diam( U ) = 2 x < R .Proof. Let Φ( re iθ ) = r (cid:96) e i(cid:96)θ denote the conformal map from the sector { re iθ : | θ | < π/ (2 (cid:96) ) } ontothe right half plane, let U + = Φ − ( D ∗ ((0 , x (cid:96) ))) . Since Φ − : C (0 ,x (cid:96) ) → C (0 ,x ) is holomorphic,by the Schwarz Lemma Φ − contracts the Poincaré metrics on these sets, and therefore U + ⊂ D ∗ ((0 , x )) . Define U − = {− z : z ∈ U + } , U = U + ∪ U − and F ( z ) = (cid:26) be − / Φ( z ) if z ∈ U + be − / Φ( − z ) if z ∈ U − . It is straightforward to check that F maps U + (resp. U − ) onto B ∗ (0 , R ) as an un-branchedcovering. (cid:3) Corollary 7.3. F ∈ F . Moreover, if c ∈ U then F c = F + c satisfies the robust separationproperty in Theorem 7.1. In particular, the kneading sequence of F c depends monotonically on c ∈ [ − β, ∞ ) .Proof. Take U as in the previous lemma and take V = B ( c, R ) . Then F c : U → V \ { c } is anunbranched covering, and since diam( U ) < R the robust separation property in Theorem 7.1 issatisfied. (cid:3) Figure 1.
The graphs of f c : [ − , → [ − , (in orange) and f c : [ − , → [ − , (in red) for various choices of c when b = 2( e(cid:96) ) /(cid:96) . For b < e(cid:96) ) /(cid:96) there exists noChebychev parameter c , and b > e(cid:96) ) /(cid:96) there exists two such parameters. Figure 2.
A bifurcation diagram. Here for c ∈ [ − , (drawn in the horizontal axis),the last 100 iterates from the set { f k (0) } k =0 are drawn (in the vertical direction). Noticethat for b = 2( e(cid:96) ) /(cid:96) + 0 . the interval [ − β c , β c ] , where β c > is the repelling fixed pointof f c (when it exists), is not invariant for parameters c ≈ − and for these parameteralmost every point is the basin of an attracting fixed point in R + . The entropy decreasesfor c ∈ [ − β, where for b = 2( e(cid:96) ) /(cid:96) + 0 . , we have β ≈ − . . Remark 7.4.
When b < e(cid:96) ) /(cid:96) , there no longer exists c so that f c is a Chebychev map,i.e. so that f c (0) = β c where β c > is a fixed point. Indeed, otherwise f c (0) = β c andtherefore since β c is a fixed point and f c is symmetric, c = f c (0) = − β c = − f c ( β c ) = − f − β c ( β c ) which implies that β c is a solution of b = 2 xe /x (cid:96) . As we have shown above, this equation hasno solution when b < e(cid:96) ) /(cid:96) . Also note that when b = 2( e(cid:96) ) /(cid:96) then c = − β = − (cid:96) /(cid:96) and dβ c dc = 1 / (1 − Df c ( β c )) = 1 / (1 −
2) = − , and therefore the positive oriented transversality of f c fails at c = − β . Families of the form f a ( x ) = af ( x ) . There are quite a few papers which ask the question:For which interval maps f , has one monotonicity of the entropy for the family x (cid:55)→ f a ( x ) , a ∈ R ?This question is subtle, as the counter examples to various conjectures show, see [39, 21, 4, 50].In this section we will obtain monotonicity and transversality for such families provided f iscontained in the large classes E , E o defined in Subsection 2.8. For convenience, let us recapitulatethe definitions of the spaces E and E o . Consider holomorphic maps f : D → V such that:(a) D, V are open sets which are symmetric w.r.t. the real line so that f ( D ) = V (b) Let I = D ∩ R then there exists c > so that I ∪ { c } is a (finite or infinite) openinterval and ∈ I , c ∈ int ( I ) . Moreover, f extends continuously to I , f ( I ) ⊂ R and lim z ∈ D,z → f ( z ) = 0 .(c) Let D + be the component of D which contains I ∩ ( c, ∞ ) , where D + might be equal to D . Then u ∈ D \ { } and v ∈ D + \ { } , v (cid:54) = u , implies u/v ∈ V .Let E be the class of maps which satisfy ( a ) , ( b ) , ( c ) and assumption ( d ) :(d) f : D → V has no singular values in V \ { , } and c > is minimal such that f has apositive local maximum at c and f ( c ) = 1 .Similarly let E o be the class of maps which satisfy ( a ) , ( b ) , ( c ) and assumption ( e ) :(e) f is odd, f : D → V has no singular values in V \ { , ± } and c > is minimal such that f has a positive local maximum at c and f ( c ) = 1 .Here, as usual, we say that v ∈ C is a singular value of a holomorphic map f : D → C if it is acritical value, or an asymptotic value where the latter means the existence of a path γ : [0 , → D so that γ ( t ) → ∂D and f ( γ ( t )) → v as t ↑ . Note that we do not require here that V ⊃ D . Using qs-rigidity, it was already shown in [41] that the topological entropy of R (cid:51) x (cid:55)→ af ( x ) is monotone a , where f ( x ) = sin( x ) or more generally f is real, unimodal and entire on thecomplex plane and satisfies a certain sector condition. Here we strengthen and generalise thisresult as follows: Theorem 7.2.
Let f be either in E or in E o . Assume that the local maximum c > is periodicfor f a ( x ) = af ( x ) where < a < b . Then the following ‘positive-oriented’ transversality propertyholds: (7.2) ddλ f qλ ( c ) | λ = a Df q − a ( f a ( c )) > . (A similar statement holds when c is pre-periodic for f a .) In particular, the kneading sequenceof the family f a ( x ) : J → R is monotone increasing.Proof. Let f ∈ E ∪ E o . Denote g ( x ) = af ( x ) and let V a = a · V . Let P = { c } , P = { c i = g i ( c ) : i ≥ } . Since < a < b , f a maps (0 , b ) into itself, and so P ⊂ (0 , b ) . We may also assumethat g ( c ) > c because otherwise q = 1 and the result is again trivial. By the assumptions, g is aholomorphic map g : D → V a , g ( P ) ⊂ P and Dg ( x ) (cid:54) = 0 for any x ∈ P \ P . In particular, g is areal marked map. For each w ∈ W := C ∗ = C \ { } , G w ( z ) := wf ( z ) is a branched covering from U := D \ { } into C . Define p ( w ) ≡ c . Then ( g, G, p ) W is a local holomorphic deformation of g . It suffices to prove that ( g, G, p ) W has the lifting property so that the Main Theorem applies.Indeed, if is an eigenvalue of A then by the Main Theorem {R ( w ) = 0 } is an open set andtherefore this critical relation holds for all parameters, which clearly is not possible.Let us first consider the case f ∈ E . In this case, w is the only critical or singular valueof G w . Given a simply connected domain ∆ (cid:51) in C , let M ∆ denote the collection of allholomorphic motions h λ of g ( P ) over (∆ , with the following property that for all λ ∈ ∆ wehave h λ ( x ) ∈ U for all x ∈ g ( P ) \ { c } and h λ ( c ) = c . Given such a holomorphic motion, foreach x ∈ g ( P ) there is a holomorphic map λ (cid:55)→ (cid:98) h λ ( x ) , λ ∈ ∆ , with (cid:98) h ( x ) = x and such that f ( (cid:98) h λ ( x )) = h λ ( g ( x )) /h λ ( g ( c )) . Indeed, for x = c , take (cid:98) h λ ( x ) ≡ c and for x ∈ g ( P ) \ { c } , we haveby property (c) that h λ ( g ( x )) /h λ ( g ( c )) ∈ V \ { , } . Note that we use here that g ( c ) ∈ D + since c < g ( c ) < b . So the existence of (cid:98) h λ follows from the fact that f : D \ f − { , } → V \ { , } isan unbranched covering. Clearly, (cid:98) h λ is a holomorphic motion in M ∆ and it is a lift of h λ over ∆ . It follows that ( g, G, p ) W has the lifting property. Indeed, if h λ is a holomorphic motion of g ( P ) over (Λ , for some domain Λ (cid:51) in C , then we can take a small disk ∆ (cid:51) such that therestriction of h λ on (∆ , is in the class M ∆ . Therefore, there exists a sequence of holomorphicmotions h ( k ) λ of g ( P ) over (∆ , such that h (0) λ = h λ and h ( k +1) λ is a lift of h ( k ) λ over ∆ for each k ≥ . If x = c then h ( k ) λ ( x ) ≡ c for each k ≥ while if x ∈ g ( P ) \ { c } , h ( k ) λ ( x ) avoids values and c . Restricting to a small disk, we conclude by Montel’s theorem that λ (cid:55)→ h ( k ) λ ( x ) is bounded.The case f ∈ E o is similar. In this case, G w has two critical or singular values w and − w ,but it has additional symmetry being an odd function. Given a simply connected domain ∆ (cid:51) in C , let M o ∆ denote the collection of all holomorphic motions h λ of g ( P ) over (∆ , with thefollowing properties: for each λ ∈ ∆ , • h λ ( x ) ∈ U for all x ∈ g ( P ) \ { c } and h λ ( c ) = c ; • h λ ( x ) (cid:54) = − h λ ( y ) for x, y ∈ g ( P ) and x (cid:54) = y .Then similar as above, we show that each h λ in M o ∆ has a lift which is again in the class M o ∆ .It follows that ( g, G, p ) W has the lifting property. (cid:3) Let us now show that this theorem applies to a unimodal family with a flat critical point:
Lemma 7.5.
The unimodal map f : [0 , → [0 , defined by (7.3) f ( x ) = exp(2 (cid:96) ) (cid:16) − exp( − / | x − / | (cid:96) ) + exp( − (cid:96) ) (cid:17) has a holomorphic unbranched extension f : D − ∪ D + → B ∗ (1 , . Here D − , D + are domainswith D − ⊂ B ∗ (0 , / , D + ⊂ D ∗ (1 / , and B ∗ ( x, r ) is the ball with radius r centered and punctured at x . There exists (cid:96) > so that for all (cid:96) > (cid:96) , one can slightly enlarge V , D − , D + and obtain a map in E .Proof. Let us consider f as a composition of a number of maps. First z (cid:55)→ (1 / − z ) (cid:96) and z (cid:55)→ ( z − / (cid:96) map some domains U − , U + which are contained in D ∗ (0 , / and D ∗ (1 / , onto D ∗ (0 , / (cid:96) ) when (cid:96) ≥ . Next z (cid:55)→ − /z maps D ∗ (0 , / (cid:96) ) onto a half-plane Re ( z ) ≤ − (cid:96) .Then z (cid:55)→ exp( z ) maps this half-plane onto the punctured disc B ∗ (0 , exp( − (cid:96) )) centered at and with radius exp( − (cid:96) ) (and with a puncture at ). Applying the translation z (cid:55)→ exp( − (cid:96) ) to this punctured disc we obtain the punctured disc centered at B ∗ (exp( − (cid:96) ) , exp( − (cid:96) )) . Thenmultiplying this disc by exp(2 (cid:96) ) shows that f maps U − , U + onto B ∗ (1 , . (Note that this finalpunctured disc touches the imaginary axis.) Since is a repelling fixed point of f with multiplier > , and U − , U + are close to the intervals (0 , / and (1 / , when (cid:96) is large, we can enlargethe domain and range, and obtain a map as in (a)-(d). (cid:3) Figure 3.
A bifurcation diagram for the family f λ ( x ) = λf ( x ) , λ ∈ [0 , where f isas formula (7.3). Note that this unimodal family with a flat critical point is monotone. Application to families with one free critical point
Let us apply the method along a curve in parameter space corresponding to where some ν -parameter family of maps G w has ν − critical relationships.Let us say that ( G, p ) W is a partially marked family of maps if(1) W is an open connected subset of C ν and U is an open subset of C ;(2) p = ( p , p , . . . , p ν ) : W → C ν is a holomorphic map and so that all coordinates of p ( w ) are distinct. Let P ,w = { p ( w ) , . . . , p ν ( w ) } .(3) G : ( w, z ) ∈ W × U (cid:55)→ G w ( z ) ∈ C is a holomorphic map and DG w ( z ) (cid:54) = 0 for each w ∈ W and z ∈ U .(4) associated to each j = 1 , . . . , ν − there exists a positive integer q j so that G kw ( w j ) ∈ U for k = 1 , . . . , q j − and w = ( w , · · · , w ν − , w ν ) ∈ W .Assume that the family map G w is real. That is, assume that the properties defined belowDefinition 2.3 hold: for any w = ( w , w , . . . , w ν ) ∈ W , z ∈ U and j = 1 , , . . . , ν , we have w = ( w , w , . . . , w ν ) ∈ W , z ∈ U , and G w ( z ) = G w ( z ) , and p j ( w ) = p j ( w ) . Choose for each j = 1 , . . . , ν − either µ ( j ) ∈ { , , . . . , ν } or ≤ l j < q j . Given this choice,let L be the set of w = ( w , · · · , w ν ) ∈ W for which the following hold:(5) if µ ( j ) is defined then G q j − w ( w j ) = p µ ( j ) ( w ) and G kw ( w j ) (cid:54)∈ P ,w for each ≤ k < q j ;(6) if l j is defined then G q j − w ( w j ) = G l j − w ( w j ) and G kw ( w j ) (cid:54)∈ P ,w for all ≤ k ≤ q j − .Relabelling these points w , . . . , w ν − , we assume that there is r such that the first alternativehappens for all ≤ j ≤ r and the second alternative happens for r < j ≤ ν − . Remark 8.1.
So for each w ∈ L , G w has ν − critical relations which start with p ( w ) , . . . , p ν − ( w ) .Hence the terminology of partially marked family of maps. Define for w ∈ W , for ≤ j ≤ r ,(8.1) R j ( w ) = G q j − w ( w j ) − p µ ( j ) ( w ) and for r < j ≤ ν − ,(8.2) R j ( w ) = G q j − w ( w j ) − G l j − w ( w j ) , where w = ( w j ) νj =1 . Then L is precisely the set L = { w ∈ W ; R ( w ) = · · · = R ν − ( w ) = 0 } . Let L ∗ be a maximal connected subset of L ∩ R ν such that for each w ∈ L ∗ , the ν × ( ν − matrix(8.3) V w := (cid:34) DG q − w ( w ) ∇ R ( w ) , . . . , DG q ν − − w ( w ν − ) ∇ R ν − ( w ) (cid:35) has rank ν − . Here ∇ R i ( w ) is the gradient of R i . By the implicit function theorem, L ∗ is areal analytic curve.Now, let us assume that for some c = ( c , , . . . , c ,ν ) ∈ L ∗ , G c has an additional criticalrelation starting with p ν ( w ) , i.e., g = G c is a marked map and G extends to a local holomorphicdeformation of g : there is a neighborhood W c ⊂ W of c , an open set U c ⊃ U such that G : W c × U → W c × C extends to a holomorphic map G : ( w, z ) ∈ W c × U c (cid:55)→ ( w, G w ( z )) ∈ W c × C and DG w | U c (cid:54) = 0 for each w ∈ W c and there exists either µ ( ν ) ∈ { , , . . . , ν } or l ν < q ν so that either • if µ ( ν ) is defined then g q ν ( p ν ( c )) = p µ ( ν ) ( c ) , g k ( c ,ν ) (cid:54)∈ P , c for each ≤ k < q ν thendefine R ν ( w ) = G q ν − w ( w ν ) − p µ ( ν ) ( w ) • if l ν exists then g q ν c ( p ν ( c )) = g l ν ( p ν ( c )) and g k ( w ν ) (cid:54)∈ P , c for all ≤ k ≤ q ν , then wedefine R ν ( w ) = G q ν − w ( w ν ) − G l ν − w ( w ν ) . Notice that R ν is only defined in a small neighbourhood W c of c .The following theorem gives a condition implying that along the curve L ∗ all bifurcations arein the same direction. Theorem 8.1.
For each w ∈ L ∗ , define E w ∈ T w C ν to be the unique unit vector in R ν orthogonalto the range of the matrix V w and so that det (cid:34) DG q − w ( w ) ∇ R ( w ) , . . . , DG q ν − − w ( w ν − ) ∇ R ν − ( w ) , E w (cid:35) > . Then • E w is a tangent vector to L ∗ at w and L ∗ (cid:51) w (cid:55)→ E w is real analytic. In particular, E w defines an orientation on the entire curve L ∗ which we will call ‘positive’. • If for some c ∈ L ∗ the corresponding map g = G c is a marked map as above and the pos-itively oriented transversality property (2.3) holds for the local holomorphic deformation ( g, G, p ) W c , then Dg q ν − ( c ,ν ) ∇ E R ν ( c ) > where Dg q ν − ( c ,ν ) is the spatial derivative, and ∇ E R ν ( c ) is the derivative in the direc-tion of the tangent vector E = E c of L ∗ at c . -1.5 -1 -0.5 0 0.5 1 1.5 a -3-2-10123 b f a,b3 (a)=af a,b3 (-a)=-a a -0.5-0.4-0.3-0.2-0.100.10.20.30.40.5 b f a,b3 (a)=af a,b3 (-a)=-a Figure 4.
The sets Γ ( ± a ) = { ( a, b ); f a,b ( ± a ) = ± a } for the cubic family f a,b = x − a x + b (with critical points ± a ). The two curves in the left panel crossing at (0 , correspond to the sets where one of the critical points of f a,b is a fixed point, i.e. where { f a,b ( ± a ) = ± a } . The two points marked with ∗ in the magnification on the right arewhere the two critical points ± a lie on the same orbit. By Theorem 8.2, the topologicalentropy is monotone on the components of Γ ( ± a ) minus these points. This informationwas already obtained by Milnor and Tresser in [37] but here we derive it from the localmethods derived in this paper. Proof.
Let A = [ A , . . . , A ν ] be the ν × ν matrix with the first ( ν − -columns equal to thecolumns of V c and the last column equal to DG q ν − c ( c ,ν ) ∇ R ν ( c ) . Note that the determinantof this matrix is positive by the positive oriented transversality condition (2.3). There exists λ so that A ν = λE c + v where v is in the range of the matrix V c . So det( A , . . . , A ν − , A ν ) =det( A , . . . , A ν − , λE c + v ) = λ det( A , . . . , A n − , E c ) . Since the first and the last determinantsare positive, we have λ > and so A ν · E c > . This is precisely the expression claimed to bepositive in the theorem. (cid:3) Remark 8.2.
Applying this theorem to the setting of a family of globally defined real analyticmaps, we obtain monotonicity of entropy along such curves L ∗ . This holds because the topologicalentropy is equal to the growth rate of the number of laps for continuous piecewise monotoneinterval or circle maps, [36] . Application to ‘bone’ curves in the space of real cubic maps.
For every ( a, b ) ∈ Σ := R \ { a = 0 } , let f a,b ( x ) = x − a x + b . Then f a,b has critical points ± a . It is clearthat for any ( a, b ) ∈ C \ { a = 0 } , f a,b is locally parametrized by its (different) critical values w , = ± a + b . For q > consider a connected component L q of the set { ( a, b ) ∈ Σ : f qa,b ( a ) ∈ {± a } , f ka,b ( a ) / ∈ {± a } , k = 1 , · · · , q − } . By [22], the corresponding × matrix (8.3) has rank one and hence L q = L q ∗ is a simple smoothcurve. In fact, the positively oriented transversality property holds for any critically-finite f a,b ;this follows similar to the proof of Theorem 7.1 of the next Section (see [24] for a general resultthough). By Theorem 8.1 we have a positive orientation on L q = L ∗ q and the entropy increasesor decreases along this curve as mentioned in Remark 8.2.For q > consider a connected component Γ q of the set { ( a, b ) ∈ Σ : f qa,b ( a ) = a } which was called a bone in [37]. The next theorem proves a crucial property of this set, which wasderived in [37] using global considerations (including Thurston rigidity for postcriticallly finitemaps). Here we will derive this property from positive transversality. Theorem 8.2 (Properties of bones) . Assume that for some (˜ a, ˜ b ) ∈ Γ q the integer q > isminimal so that f q ˜ a, ˜ b (˜ a ) = ˜ a . Then for all ( a, b ) ∈ Γ q one has that f ia,b ( a ) (cid:54) = a for all < i < q .Moreover,(1) there exists at most one ( a ∗ , b ∗ ) ∈ Γ q so that f ia ∗ ,b ∗ ( a ∗ ) = − a ∗ for some ≤ i ≤ q .(2) the kneading sequence of f a,b is monotone on each of the components of Γ q \ { ( a ∗ , b ∗ ) } ;more precisely, it is non-decreasing on one component and non-increasing on the othercomponent.Proof. That f ia,b ( a ) (cid:54) = a for all < i < q , ( a, b ) ∈ Γ q follows from the implicit function theorembecause the multiplier of this q -periodic orbit is not equal to . It is well known that Γ q is asmooth curve (this also follows for example from [22]). The curve Γ q is a component of the zeroset of ˜ R ( a, b ) = f qa,b ( a ) − a. As remarked, the critical values w = ( w , w ) are local parameters along the curve Γ q and wedefine a direction on the curve Γ q by the tangent vector V ( a,b ) = ( − ∂ ˜ R∂w , ∂ ˜ R∂w ) .Assume that for some ( a ∗ , b ∗ ) ∈ Γ q the orbit of a ∗ contains the other critical point, i.e. assumethat f ia ∗ ,b ∗ ( a ∗ ) = − a ∗ for some < i < q . The idea of the proof below is as follows. We will showthat as the point ( a, b ) ∈ Γ q passes through ( a ∗ , b ∗ ) , the point − a crosses f ia,b ( a ) in a directionwhich depends only on the sign of ∆ i ( a ∗ , b ∗ ) where(8.4) ∆ i ( a, b ) := (cid:89) ≤ k As in the proof of Theorem 8.1 we obtain(8.5) i ( a ∗ , b ∗ ) D V ( a ∗ ,b ∗ ) R ( a ∗ , b ∗ ) > where D V ( a ∗ ,b ∗ ) stands for the directional derivative of R in the direction V ( a ∗ ,b ∗ ) . To be definite,let us consider the case that(8.6) ∆ i ( a ∗ , b ∗ ) < . This implies that(8.7) D V ( a ∗ ,b ∗ ) R ( a ∗ , b ∗ ) < and so the derivative of(8.8) Γ q (cid:51) ( a, b ) (cid:55)→ f ia,b ( a ) − ( − a ) is negative at ( a ∗ , b ∗ ) . By contradiction, assume that there exists another parameter ( a • , b • ) ∈ Γ q which is the nearestto the right of ( a ∗ , b ∗ ) for which there exists < j < q so that f ja • ,b • ( a • ) = − a • . In what followswe use that the ordering of the points a, . . . , f q − a,b ( a ) in R does not change along the curve Γ q .Let Γ • q be the open arc between ( a ∗ , b ∗ ) and ( a • , b • ) . Notice that because of (8.6)(8.9) ∆ i ( a, b ) < for all ( a, b ) ∈ Γ • q . Observe that j (cid:54) = i because when j = i then (8.9) holds on the closureof Γ • q and so D V ( a • ,b • ) R ( a • , b • ) < . Therefore (8.8) also holds at ( a • , b • ) which is clearly acontradiction. Therefore, j (cid:54) = i and(8.10) f ia,b ( a ) < − a < f ja,b ( a ) along the open arc Γ • q and there are no points of the orbit of a between f ia,b ( a ) , f ja,b ( a ) . The signof(8.11) ∆ j ( a, b ) := Df j − a,b ( f a,b ( a )) Df q − j − a,b ( f a,b ( − a )) is constant for ( a, b ) near ( a • , b • ) . Therefore for ( a, b ) ∈ Γ • q ,sign ∆ j ( a • , b • ) = sign ∆ j ( a, b ) = sign (cid:34) ∆ i ( a, b ) Df a,b ( f ia,b ( a )) Df a,b ( f ja,b ( a )) (cid:35) . Because of (8.9) we therefore havesign ∆ j ( a • , b • ) = − sign (cid:34) Df a,b ( f ia,b ( a )) Df a,b ( f ja,b ( a )) (cid:35) . The key point is that the sign of the ratio in the r.h.s. of this expression is negative because − a is a folding critical point and because of (8.10). It follows that ∆ j ( a • , b • ) > and so arguing asbefore the derivative of(8.12) Γ q (cid:51) ( a, b ) (cid:55)→ f ja,b ( a ) − ( − a ) is positive at ( a • , b • ) . But by (8.10) we have that f ja,b ( a ) − ( − a ) > on Γ • q . This and (8.12) imply that f ja • ,b • ( a • ) − ( − a • ) > which is a contradiction.The 2nd assertion follows immediately from Theorem 8.1 and Remark 8.2. (cid:3) Remark 8.3.
The proof of the previous theorem can also be applied to the setting of polynomialsof higher degrees.
Appendix A. The family f c ( x ) = | x | (cid:96) ± + c with (cid:96) ± > large In this section we obtain monotonicity for unimodal (not necessary symmetric!) maps in thepresence of critical points of large non-integer order, but only if not too many points in thecritical orbit are in the orientation reversing branch.A.1.
Unimodal family with high degrees.Theorem A.1.
Fix real numbers (cid:96) − , (cid:96) + ≥ and consider the family of unimodal maps f c = f c,(cid:96) − ,(cid:96) + where f c ( x ) = (cid:26) | x | (cid:96) − + c if x ≤ | x | (cid:96) + + c if x ≥ . For any integer L ≥ there exists (cid:96) > so that for any q ≥ and any periodic kneadingsequence i = i i · · · ∈ {− , , } Z + of period q so that { ≤ j < q ; i j = − } ≤ L, and any pair (cid:96) − , (cid:96) + ≥ (cid:96) there is at most one c ∈ R for which the kneading sequence of f c isequal to i . Moreover, (A.1) q − (cid:88) n =0 Df nc ( c ) > . Notations.
As usual, for any three distinct point o, a, b ∈ C , let ∠ aob denote the angle in [0 , π ] which is formed by the rays oa and ob . We shall often use the following obvious relation:for any distinct four points o, a, b, c , ∠ aob + ∠ boc ≥ ∠ aoc. For θ ∈ (0 , π ) , let D θ = { z ∈ C \ { , } : ∠ z > π − θ } and let S θ = { re it : t ∈ ( − θ, θ ) } . For < t < , we shall only consider z t in the case z (cid:54)∈ ( −∞ , and z t is understood as theholomorphic branch with t = 1 .Let us fix a map f = f c,(cid:96) − ,(cid:96) + with a periodic critical point of period q and let P = { f n (0) : n ≥ } . So P is a forward invaraint finite set. Denote (cid:96) = min { (cid:96) − , (cid:96) + } . Definition A.1.
A holomorphic motion h λ of P over (Ω , , is called θ -regular if(A1). For a ∈ P , h λ ( a ) ∈ S θ/(cid:96) , if a > and h λ ( a ) ∈ − S θ/(cid:96) , if a < (A2). For a, b ∈ P , | a | > | b | > and ab > , h λ ( b ) h λ ( a ) ∈ D θ . Given a θ -regular holomorphic motion h λ of P over Ω , with θ ∈ (0 , π ) , one can define anotherholomorphic motion ˜ h λ of P over the same domain Ω as follows: ˜ h λ (0) = 0 ; for a ∈ P with a > , ˜ h λ ( a ) = ( h λ ( f ( a )) − h λ ( f (0))) /(cid:96) + ; for a ∈ P with a < , define ˜ h λ ( a ) = − ( h λ ( f ( a )) − h λ ( f (0))) /(cid:96) − . The new holomorphic motion is called the lift of h λ which clearly satisfies the condition (A1),but not necessarily (A2) in general. Main Lemma.
There is (cid:96) depending only on the number L such that for any (cid:96) ≥ (cid:96) and each θ small enough, the following holds: If { ≤ j < q ; i j = − } ≤ L and if a θ -regular motion canbe successively lifted q − times and all these successive lifts are θ -regular, then the q -th lift ofthe holomorphic motion is θ/ -regular.Proof of Theorem A.1. Given L , choose (cid:96) as in the Main Lemma. It is enough to prove (A.1)provided (cid:96) ≥ (cid:96) . Consider a local holomorphic deformation ( f c , f w , p ) W where W ⊂ C is a smallneighbourhood of c , f w = f c + ( w − c ) and p = 0 . Let h λ be a holomorphic motion of P over (∆ , . Let us fix θ > small enough. Restricting h λ to a smaller domain ∆ ε , we may assumethat h λ is θ -regular and that h λ can be lifted successively for q times. Therefore by the MainLemma, we obtain a sequence of holomorphic motions h ( n ) λ of P over (∆ ε , , such that h (0) λ = h λ and h ( n +1) λ is the lift of h ( n ) λ and such that h ( n ) λ ( x ) ∈ ± S θ n for all n and all x ∈ P where θ n → as n → ∞ . Thus ( f c , f w , p ) W has the lifting property and by the Main Theorem, the transversalitycondition (A.1) holds. Alternatively, the uniqueness of c follows directly from the Main Lemma. Indeed, let ˜ f = f ˜ c be a map with the same kneading sequence as f c . Then one can define a real holomorphic motion h λ over some domain Ω (cid:51) , such that h λ ( f n (0)) = ˜ f n (0) for λ = 1 . As above, for i > let h ( i ) λ be the lift of h ( i − λ . As we have just shown, h ( i ) λ ( c ) is contained in the sector − S θ n with θ n → , this sequence of functions λ → h ( i ) λ ( c ) has to converge to a constant function. Since byconstruction of the lifts ˜ c = h ( n )1 ( c ) for each n ≥ we conclude that ˜ c = c . (cid:3) A.2.
Proof of the Main Lemma.Lemma A.2.
For any θ ∈ (0 , π ) and < t < , if z ∈ D θ then z t ∈ D θ .Proof. This is a well-known consequence of the Schwarz lemma, due to Sullivan. (cid:3)
When ∠ z is much smaller than ∠ z , we have the following improved estimate. Lemma A.3.
For any ε > , there is δ > such that the following holds. For z ∈ D θ with θ ∈ (0 , π/ and ∠ z < δθ and for any < t < , we have ∠ z t < εθ. Proof.
Write z = re iα where r > and α ∈ (0 , θ ) and write α (cid:48) = tα and β (cid:48) = ∠ z t . Byassumption, α + β ≤ θ . By the sine theorem, r = sin β sin( α + β ) and r t = sin β (cid:48) sin( tα + β (cid:48) ) . If α + β < εθ then by Lemma A.2, α (cid:48) + β (cid:48) ≤ α + β < εθ. Assume now α + β ≥ εθ . Let K > be a large constant such that tK t − < ε for any < t < . Assume β < δθ for δ small. Then r < /K . Thus tan β (cid:48) = r t sin tα − r t cos tα ≤ tr t − r t α ≤ tK t − α < εθ. (cid:3) Lemma A.4.
Let ϕ λ be a θ -regular motion with θ ∈ (0 , π/ and let ψ λ be its lift. For x, y ∈ P so that xy ≥ let x λ = ψ λ ( x ) , y λ = ψ λ ( y ) , u λ = ϕ λ ( f ( x )) , v λ = ϕ λ ( f ( y )) and c λ = ϕ λ ( f (0)) .For any ε > there is (cid:96) and δ > such that if (cid:96) > (cid:96) then the following hold.(1) If f ( x ) ≤ ≤ f ( y ) then ∠ x λ y λ ≥ π − εθ for all λ .(2) Let < f ( x ) < f ( y ) . Then (i) ∠ x λ y λ ≥ ∠ u λ v λ − θ(cid:96) . If, moreover, c λ ∈ − S θ and u λ , v λ ∈ S θ for some θ ∈ (0 , θ/(cid:96) ] then (ii) x λ , y λ ∈ ± S θ /(cid:96) and ∠ x λ y λ ≥ ∠ u λ v λ − θ .(3) Suppose f ( x ) < f ( y ) < and α = π − min( ∠ c λ v λ , ∠ u λ v λ < δθ. Then ∠ x λ y λ ≥ π − εθ. Proof.
Note that (cid:52) xy is the image of (cid:52) c λ u λ v λ under an appropriate branch of z (cid:55)→ ( z − c λ ) t .Since ∠ xoy < θ/(cid:96) , an upper bound on ∠ oyx implies a lower bound on ∠ oxy .(1) In this case, we have u λ ∈ − S θ/(cid:96) and v λ ∈ S θ/(cid:96) , so ∠ u λ v λ ≤ θ/(cid:96), and ∠ v λ u λ ≤ θ/(cid:96). In particular, ∠ c λ u λ v λ ≥ ∠ c λ u λ − ∠ u λ v λ ≥ π − θ − θ/(cid:96) ≥ π − θ. By Lemma A.3, the statement follows. (2) In this case, ∠ c λ u λ v λ ≥ ∠ u λ v λ − ∠ u λ c λ ≥ ∠ u λ v λ − θ/(cid:96). Thus by Lemma A.2, the conclusion (i) follows; (ii) is similar.(3) In this case, ∠ c λ u λ v λ ≥ ∠ c λ u λ − ∠ v λ u λ ≥ π − θ − α ≥ π − θ and ∠ c λ v λ u λ ≤ π − ( ∠ c λ v λ ∠ v λ u λ ) ≤ α. So the conclusion follows from Lemma A.3. (cid:3)
Now suppose that we have a sequence of θ -regular holomorphic motions h ( i ) λ of P , i =0 , , . . . , q − over the same marked domain (Ω , , such that h ( i ) λ is a lift of h ( i − λ for all ≤ i < q . Then h ( q ) λ , lift of h ( q − λ is well-defined and satisfies the condition (A1) with the sameconstant θ . For each ≤ i ≤ q , λ ∈ Ω and x, y ∈ P so that < | x | < | y | and xy > , let θ iλ ( x, y ) = π − inf { ∠ h ( i ) λ ( z ) h ( i ) λ ( z ) : z , z ∈ P, < | z | ≤ | x | < | y | ≤ | z | , xz > , xz > }≥ π − ∠ h ( i ) λ ( x ) h ( i ) λ ( y ) . Furthermore, given any x, y ∈ P , xy > (but not necessarily | x | < | y | ), denote ˆ θ i ( x, y ) = θ i ( x ∧ y, x ∨ y ) where x ∧ y = x/ | x | min( | x | , | y | ) and x ∨ y = x/ | x | max( | x | , | y | ) . Lemma A.5.
Consider ≤ i < q , x, y ∈ P where xy > and λ ∈ Ω . For any ε > there is δ > and (cid:96) > such that if (cid:96) ≥ (cid:96) , then the following hold.(1) If f ( x ) ≤ ≤ f ( y ) then ˆ θ i +1 λ ( x, y ) ≤ εθ. (2) Let r ≥ be such that i + r ≤ q . If < f j ( x ) < f j ( y ) for all ≤ j ≤ r , then ˆ θ i + rλ ( x, y ) ≤ ˆ θ iλ ( f r ( x ) , f r ( y )) + εθ. (3) If f ( x ) < f ( y ) < and ˆ θ iλ ( f ( x ) , f ( y )) < δθ , then ˆ θ i +1 λ ( x, y ) ≤ εθ, ˆ θ iλ ( f ( x ) , f ( y ))) . Proof.
Note that f ( x ) < f ( y ) implies | x | < | y | .(1) For each < | z | ≤ | x | < | y | ≤ | z | as in the definition of θ iλ ( x, y ) we have f ( z ) ≤ and f ( z ) ≥ . So by Lemma A.4 (1), (applying to ϕ = h ( i ) and ψ = h ( i +1) ), ∠ h ( i +1) λ ( z ) h ( i +1) λ ( z ) ≥ π − εθ . Thus the statement holds.(2) Consider < | z | ≤ | x | < | y | ≤ | z | so that z z > . Then f ( z ) > . If f ( z ) ≤ , thenby Lemma A.4 (1), ∠ h ( i + r ) λ ( z ) h ( i + r ) λ ( z ) ≥ π − εθ . Assume f ( z ) > and let r ∈ { , · · · , r } be maximal such that < f j ( z ) ≤ f j ( x ) < f j ( y ) ≤ f j ( z ) for all ≤ j ≤ r . Notice that then < f r ( z ) ≤ f r − ( f (0)) < f r − ( f (0)) < · · · < f (0) . Let us show that for all k ∈ { , · · · , r − } ,(A.2) h ( k + i + r − r ) λ ( f (0)) ∈ − S θ/(cid:96) k +1 , and(A.3) h ( k + i + r − r ) λ ( f r − k ( z )) , h ( k + i + r − r ) λ ( f r − k ( z )) ∈ S θ/(cid:96) k +1 . Indeed, this holds for k = 0 as h ( i + r − r ) λ is θ -regular. Now, for ≤ k ≤ r − , (A.2)-(A.3) followsby a successive application of the second part of Lemma A.4 (2). This proves (A.2)-(A.3). Inturn, using (A.2)-(A.3) and again applying successively Lemma A.4 (2), ∠ h ( i + r ) λ ( z ) h ( i + r ) λ ( z ) > ∠ h ( i + r − r ) λ ( f r ( z )) h ( i + r − r ) λ ( f r ( z )) − ∞ (cid:88) k =0 θ(cid:96) k +1 = ∠ h ( i + r − r ) λ ( f r ( z )) h ( i + r − r ) λ ( f r ( z )) − θ(cid:96) − . Consider two cases. If r < r , then f r +1 ( z ) ≤ and f r +1 ( z ) > and by Lemma A.4 (1), ∠ h ( i + r − r ) λ ( f r ( z )) h ( i + r − r ) λ ( f r ( z )) ≥ π − εθ for any (cid:96) large enough. If r = r , ∠ h ( i ) λ ( f r ( z )) h ( i ) λ ( f r ( z )) ≥ π − θ iλ ( f r ( x ) , f r ( y )) . In any case, ∠ h ( i + r ) λ ( z ) h ( i + r ) λ ( z ) > π − θ iλ ( f r ( x ) , f r ( y )) − εθ provided (cid:96) is large enough. Thus the statement holds.(3) Notice that in this case ˆ θ iλ ( f ( x ) , f ( y )) = θ iλ ( f ( y ) , f ( x )) . Consider < | z | ≤ | x | < | y | ≤| z | so that z z > . If f ( z ) > then by Lemma A.4 (1), ∠ h ( i +1) λ ( z ) h ( i +1) λ ( z ) ≥ π − εθ .Assume f ( z ) < . Then > f ( z ) ≥ f ( y ) > f ( x ) > f ( z ) > c . So ∠ h ( i ) λ ( c ) h ( i ) λ ( f ( z ))0 ≥ π − θ iλ ( f ( y ) , f ( x )) and ∠ h ( i ) λ ( f ( z )) h ( i ) λ ( f ( z ))0 ≥ π − θ iλ ( f ( y ) , f ( x )) . By Lemma A.4 (3), ∠ h ( i +1) λ ( z ) h ( i +1) λ ( z ) ≥ π − θ iλ ( f ( y ) , f ( x )) , εθ ) , provided that θ iλ ( f ( y ) , f ( x )) /θ is small enough and (cid:96) is large enough. (cid:3) Completion of proof of the Main Lemma.
It is easy to check that h qλ satisfies the condition (A1)with S θ/(cid:96) replaced by S θ/(cid:96) . It remains to check that for x, y ∈ P , < | x | < | y | and xy > implies ∠ h qλ ( x ) h qλ ( y ) > π − θ/ . Since the critical point is periodic, there is a minimal integer p , less than the period q of the critical point, such that f p ([ x, y ]) (cid:51) . Let us define p − m > m > · · · > m j − > m j = 0 inductively as follows. Given m i , let m j +1 ∈ { , · · · , m j − } be the maximal so that f m j +1 ([ x, y ]) ⊂ R − if it exists and m j +1 = 0 otherwise. Note that j ≤ L + 1 . Let κ m j = ˆ θ q − m j λ ( f m j ( x ) , f m j ( y )) /θ, j = 0 , , . . . , j . Fix ε > small. Assume that (cid:96) is large. Then by Lemma A.5 (1), κ m = κ p − ≤ ε. For each < j ≤ j , by Lemma A.5 (2) and (3), κ m j +1 ≤ κ m j + 4 ε provided that κ m j is small enough and (cid:96) is large enough. Therefore, provided that (cid:96) is largeenough, we have κ < / . It follows that ∠ h ( q ) λ ( x ) h ( q ) λ ( y ) ≥ π − κ θ ≤ π − θ/ . (cid:3) Appendix B. Families without the lifting property
In this appendix we will give a few examples of families for which the lifting property doesnot hold.B.1.
Remark on the lifting property for the flat family.
Using the notations of Section 7.2let b = 2( e(cid:96) ) /(cid:96) , c = − β = − (cid:96) /(cid:96) and f = F c so that (cid:55)→ c (cid:55)→ β (cid:55)→ β by f . By Remark 7.4the transversality fails for ( f, F, p ) . (It can be also checked directly that the function R ( w ) = F w ( w ) − F w ( w ) vanished at w = c , not identically zero but R (cid:48) ( c ) = 0 .) Therefore, by the MainTheorem, this triple does not have the lifting property.The purpose of this remark is to give a more direct argument that the lifting property doesnot hold for ( f, F, p ) W . So let us assume by contradiction that ( f, F, p ) has the lifting property.Fix r > small so that the function E ( z ) := exp {− z − (cid:96) } is univalent in a disk B ( β, r ) ⊂ U + . Byassumption, given an arbitrary holomorphic motion h λ of { c, β } there exist (cid:15) > and a sequenceof holomorphic motions { h ( k ) λ } ∞ k =0 of { c, β } over D (cid:15) such that for all λ ∈ D (cid:15) and all k ≥ wehave that − h ( k ) λ ( c ) , h ( k ) λ ( β ) ∈ B ( β, r ) . Therefore, bE ( − h ( k +1) λ ( c )) + h ( k ) λ ( c ) = h ( k ) λ ( β ) ,bE ( h ( k +1) λ ( β )) + h ( k ) λ ( c ) = h ( k ) λ ( β ) and h ( k +1) λ ( c ) = − h ( k +1) λ ( β ) . Let us now choose h λ ( c ) = c − λ and h λ ( β ) = − h λ ( c ) = β + λ .Then h ( k ) λ ( c ) = − h ( k ) λ ( β ) holds for all k ≥ . Hence, for a k ( λ ) := h ( k ) λ ( β ) and all λ ∈ D (cid:15) , k ≥ , b exp {− a k +1 ( λ ) (cid:96) } = 2 a k ( λ ) . On the other hand, it is easy to check that the for function f : (0 , + ∞ ) → (0 , + ∞ ) , f ( x ) = b {− x (cid:96) } , we have: f ( β ) = β , Df ( β ) = 1 , D f ( β ) < , f : [ β, + ∞ ) → [ β, b/ is an increasing homeo-morphism so that f k ( x ) → β as k → + ∞ for all x ∈ [ β, + ∞ ) . Let U : [ β, b/ → [ β, + ∞ ) be abranch of f − such that U ( β ) = β . Since a k (0) = β and functions a k are continuous in D (cid:15) , itfollows that a k ( λ ) = U k ( a ( λ )) for all k > and all λ provided a ( λ ) = β + λ ≥ β . Fix λ ∈ (0 , (cid:15) ) so that a ( λ ) = β + λ > β .It follows that h ( k ) λ ( β ) = a k ( λ ) = U k ( a ( λ )) → + ∞ as k → ∞ , a contradiction with thedefinition of the lifting property.B.2. Spectrum of the transfer operator and linear coordinate changes of the quadraticfamily.
Consider the standard holomorphic deformation of a critically finite quadratic map ( g, G, p ) , that is, p ( w ) = 0 , G w ( z ) = z + w and g = G c is so that is periodic for g of period q ≥ .Given a function ν which is locally holomorphic in a neighborhood of a point v ∈ C \ { } sothat ν ( v ) = c , let w = ν ( v ) and ϕ ( w ) = ϕ ( ν ( v )) = ν ( v ) /v . Define G νv ( z ) = 1 ϕ ( w ) G w ( ϕ ( w ) z ) = ν ( v ) v z + v and g ν = G νv .Then ( g ν , G ν , p ) is a local holomorphic deformation of g ν . Denote by A respectively A ν thetransfer operator of ( g, G, p ) respectively of ( g ν , G ν , p ) . Note that / is always in the spectrumof A (see [26]). Proposition B.1. det(1 − ρ A ν ) = 1 − v ν (cid:48) ( v ) c − ρ det(1 − ρ A ) . Proof.
Let L ν ( z ) = ∂G νv ( z ) /∂v | v = v , v n = g nν (0) , D ν ( ρ ) = 1 + q − (cid:88) n =1 ρ n L ν ( v n )( g nν ) (cid:48) ( v ) ,D ( ρ ) = 1 + q − (cid:88) n =1 ρ n g n ) (cid:48) ( c ) . We have to varify the following identity: D ν ( ρ ) = 1 − ρ (1 − v ν (cid:48) ( v ) ν ( v ) )1 − ρ D ( ρ ) . Let us indicate its proof. We have: ( g nν ) (cid:48) ( v ) = ( g n ) (cid:48) ( c ) , v n = c n /ϕ ( c ) (in particular, ϕ ( c ) = c /v , and L ν ( v n ) = ( ν ( v ) v ) (cid:48) | v = v v n + 1 . Then the above identity turns out to be equivalent to the following one: − ρ c q − (cid:88) n =1 ρ n − c n ( g n ) (cid:48) ( c ) = D ( ρ ) which is checked directly using c n +1 − c n = c for ≤ n ≤ q − and − c q − = c . (cid:3) Corollary B.2.
The transversality of ( g ν , G ν , p ) fails if and only if ν (cid:48) ( v ) = 0 . Corollary B.3.
Let c and ν be real. Then ( g ν , G ν , p ) has the lifting property if and only if thepositive transversality property holds.Proof. By the Main Theorem, the lifting property yields the positive transversality. Conversaly,by Proposition B.1, D ν (1) > implies that the spectrum of A ν belongs to the open unit diskwhich in turn implies the lifting property. (cid:3) Appendix C. The lifting property in the setting of rational mappings
The goal of this section is to show how to apply the method developed in this paper to dealwith transversality problems in the classical case of polynomials and rational maps. It is naturalto assume that the maps involved are suitably normalized, so we shall only consider the followingsituations:(a) f is a map in P d , the family of monic centered polynomials of degree d ≥ ;(b) Z is a set with Z = 3 and Rat Zd is the family of all rational maps f of degree d suchthat f ( Z ) = Z and such that Z is disjoint from the critical orbit of f .Note that for each rational map f of degree d ≥ , it is possible to find Z , consisting of eithera cycle of period , or a fixed point and a cycle of period , such that f ∈ Rat Zd . Let U = P d incase (a) and U = Rat Zd in case (b).In case (b), we assume without loss of generality that critical points and their orbits avoidsthe point at ∞ . Let c , c , · · · , c ν be the distinct (finite) critical points of f with multiplicities µ , µ , · · · , µ ν and let v j = f ( c j ) . In the following, we fix f and let U f denote the subcollection ofmaps in U which have exactly ν critical points with multiplicity µ , µ , . . . , µ ν . It is well-knownthat U f is a complex manifold of dimension ν and the critical values are holomorphic coordinates.See for example [22]. Proposition C.1.
There is a neighbourhood W of ( v , v , · · · , v ν ) in C ν such that W (cid:51) w (cid:55)→ f w in P d (resp. Rat Zd ) is biholomorphsim from W to a neighborhood of f in U f , and a holomorhicfunction p : W → C ν , such that p j ( w ) is a critical point of f w of multiplicity µ j and w = ( f w ( p ( w )) , f w ( p ( w )) , · · · , f w ( p ν ( w ))) . Now we assume that f is critically finite. Let P = P ( f ) , P = { c , c , . . . , c ν } and let U be asmall neighborhood of P \ P . Then f , restricting to P ∪ U , is a marked map in the sense of§2.1. We shall use the same notation f for the marked map. Moreover, defining F ( w, z ) = f w ( z ) , ( f, F, p ) W is a holomorphic deformation of f in the sense of §2.2. Theorem C.1.
The holomorphic deformation ( f, F, p ) W satisfies the lifting property.Proof. Let h (0) λ be an arbitrary holomorphic motion of f ( P ) over ( D , . By Bers-Royden [3],there exists ε > such that h (0) λ , λ ∈ D ε , extends to a holomorphic motion of C over ( D ε , which satisfies the following: in case (a), for | z | large enough, h (0) λ ( z ) is holomorphic in z and h (0) λ ( z ) = z + o (1) near infinity, moreover, fix a big enough disk V such that f − ( V ) ⊂ V andsuch that the complex dilatation µ (0) λ of h (0) λ is supported in V for all λ ∈ D ε , and in case (b), h λ ( z ) = z for all z ∈ Z . Moreover, the complex dilatation µ (0) λ of h (0) λ depends holomorphicallyin λ . Define µ ( k ) λ = ( f k ) ∗ ( µ (0) λ ) for each k ≥ (here f is considered as a globally defined map)and let h ( k ) λ denote the unique qc map with complex dilatation µ ( k ) λ and such that the followingholds: in case (a), h ( k ) λ ( z ) = z + o (1) near infinity; and in case (b), h ( k ) λ ( z ) = z for all z ∈ Z .Then by the Measurable Riemann Mapping Theorem, h ( k ) λ is a holomorphic motion of C over ( D ε , . Let us show that for each k ≥ , h ( k +1) λ , restricting to f ( P ) , is a lift of h ( k ) λ , restrictingto f ( P ) , with respect to ( f, F, p ) W . This amounts to prove the following: Claim.
For | λ | small enough, we have h ( k ) λ ◦ f ◦ ( h ( k +1) λ ) − = f ( h ( k ) λ ( v ) ,h ( k ) λ ( v ) , ··· ,h ( k ) λ ( v ν )) . Proof of Claim: for each λ ∈ D ε , the complex dilatation of h ( k +1) λ is the f -pull back of that of h ( k ) λ , and therefore the function g λ := h ( k ) λ ◦ f ◦ ( h ( k +1) λ ) − is holomorphic in C . It is a branchedcovering of degree d , so it is either a polynomial or a rational function of degree d . By thenormalization of both h ( j ) λ , j = k, k + 1 , in case (a), g λ ∈ U , and hence g ∈ U f . Clearly thecritical values of g λ are h ( k ) λ ( v i ) . The claim follows.To complete the proof, notice that in the case (a), µ ( k ) λ are supported in f − k ( V ) ⊂ V and bycompactness of K-qc maps the conclusion follows. (cid:3) Corollary C.2.
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