aa r X i v : . [ m a t h . D S ] J a n RESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS
HONGMING NIE
Abstract.
Suppose { f t } is an analytic one-parameter family of rational mapsdefined over a non-Archimedean field K . We prove a finiteness theorem forthe set of rescalings for { f t } . This complements results of J. Kiwi. Introduction
Let K be an algebraically closed field. For d ≥ , let Rat d ( K ) be the space of de-gree d rational maps over K , thought of as dynamical systems on P ( K ) . The group PGL ( K ) acts on Rat d ( K ) by conjugation. The moduli space of degree d rationalmaps on P ( K ) is the quotient space M d ( K ) := Rat d ( K ) / PGL ( K ) . Milnor [19]considered the moduli space M ( C ) of quadratic complex rational maps and gavea dynamically natural compactification of M ( C ) . Then using geometric invarianttheory, Silverman [21, 22] compactified the moduli space M d ( K ) in general. De-Marco [8] also considered different compactifications of the moduli space M ( C ) .To study the dynamics of complex rational maps approaching the boundary of Rat d ( C ) (or M d ( C ) ), Kiwi [18] considered rescaling limits for a holomorphic family { f t } (resp. a sequence { f n } ) in Rat d ( C ) . These arise as limits M − t ◦ f qt ◦ M t → g (resp. M − n ◦ f qn ◦ M n → g ) of rescaled iterates where the convergence is locallyuniform outside some finite subset of P ( C ) . By regarding a holomorphic familyas a rational map with coefficients in the field of formal Puiseux series, and bystudying its induced action on the corresponding Berkovich space, Kiwi proved forany given holomorphic one-parameter family of degree d ≥ rational maps, thereare at most d − dynamically independent rescalings such that the correspondingrescaling limits are not postcritically finite [18, Theorem 1, Theorem 2]. Later,Arfeux [1] proved the same results using the Deligne-Mumford compactifications ofthe moduli spaces of the stable punctured spheres.An algebraically closed complete valued field is isomorphic to either the field ofcomplex numbers C or a non-Archimedean field [9]. Our main result translatesKiwi’s finiteness result to non-Archimedean algebraically closed fields, subject to anatural tameness hypothesis. We now set up the statement.Throughout this paper, K will denote an algebraically closed field which is com-plete with respect to a nontrivial non-Archimedean absolute value | · | K . Let φ ( z ) ∈ K ( z ) be a rational map. We can write φ = φ ◦ φ , where φ is a sep-arable rational map and φ ( z ) = z p j for some j ≥ if the field K has positivecharacteristic p > or φ ( z ) = z if the field K has characteristic zero. The rationalmap φ is called the separable part of φ . The degree of φ is called the nontrivial Mathematics Subject Classification.
Primary 37P45, 37P50.
Key words and phrases.
Rescaling limits, non-Archimedean dynamics, Berkovich spaces. degree of φ and the preimages of critical points of φ under φ are called the non-trivial critical points of φ . We say rational map φ ∈ K ( z ) is postcritically finite atnontrivial critical points if each nontrivial critical point of φ has a finite forwardorbit; equivalently, each critical value of φ has a finite forward orbit under φ .As in Kiwi’s study of degenerating rational maps over C , we now study families φ t approaching boundary of Rat d ( K ) (or M d ( K ) ). Since the field K is not locallycompact with respective to the absolute value | · | K , the definition of rescaling limitsin [18] needs to be slightly modified. The non-Archimedean property turns out tomake pointwise convergence suitable; see Definition 2.4 and Proposition 3.5. Forinstance, let f t ( z ) = ( z + t ) /z ∈ K ( z ) , then, as t → , f t ( z ) converges to z pointwise on P ( K ) \ { } , but f t (0) = ∞ for all t = 0 . Since the field of formalPuiseux series over K is not algebraically closed if char K = p > , we workon the field L of Hahn series over K . For Puiseux series and Hahn series, werefer [13, 15–17]. For an analytic family { f t } ⊂ Rat d ( K ) , we can associate to { f t } a rational map f : P ( L ) → P ( L ) . The space P ( L ) is naturally a subset of thecorresponding Berkovich space P ( L ) ; see [3, 14] for details. The rational map f induces a map on P ( L ) extending its action on P ( L ) , so we also use notation f for the induced map. Let R f be the Berkovich ramification locus, that is the setof points in P ( L ) such that the local degrees of f at these points are at least , i.e R f = { ξ ∈ P ( L ) : deg ξ f ≥ } . It is a closed subset of P ( L ) with noisolated points and has at most deg f − connected components [10, Theorem A],each of which has tree structure. Following Trucco [23], we say the rational map f : P ( L ) → P ( L ) is tame if R f has only finitely many points with valence atleast in R f .We will prove Theorem 1.1.
Let { f t ( z ) } ⊂ K ( z ) be an analytic family of rational maps of non-trivial degree d ≥ and let f be the separable part of the associated rational map f of { f t } . Assume f is tame. Then there are at most d − pairwise dynamicallyindependent rescalings for { f t } such that the corresponding rescaling limits are notpostcritically finite at nontrivial critical points. In section 6, we give some examples of analytic families with rescaling limits thatare not postcritically finite at nontrivial critical points.The tameness hypothesis of the separable part f is needed in order to prove anon-Archimedean Rolle’s theorem in positive characteristic, see Lemma 5.1. If K has characteristic zero or positive characteristic p > deg f , then the rational map f : P ( L ) → P ( L ) is tame [10, Corollary 6.6]. Thus Corollary 1.2.
Suppose the field K has characteristic zero. Let { f t ( z ) } ⊂ K ( z ) be an analytic family of degree d ≥ rational maps. Then there are at most d − pairwise dynamically independent rescalings for { f t } such that the correspondingrescaling limits are not postcritically finite. Outline.
In section 2, we recall the relevant backgrounds of Berkovich space anddefine the rescaling limits for an analytic family of rational maps over K . Thegoal of section 3 is to discuss the reduction map and show the relations betweenreductions and rescaling limits. Section 4 is devoted to restating Kiwi’s resultswhich are still true for the case when K has characteristic zero. Finally, we proveTheorem 1.1 in section 5 and give examples to illustrate it in section 6. ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 3 Preliminaries
Non-Archimedean fields.
For the field K , let | K × | K ⊂ (0 , ∞ ) be the set ofabsolute values attained by nonzero elements of K , which is called the value groupof K . Then | K × | K is dense in (0 , ∞ ) since K is algebraically closed, and hence K can not be locally compact. Let O K = { z ∈ K : | z | K ≤ } be the ring of integersof K and let M K = { z ∈ K : | z | K < } be the unique maximal ideal of O K . Let k = O K / M K be the residue field. Note if char K = p > then char k = p , but if char K = 0 , then k could have any characteristic. For instance, for a prime number p ≥ , if K is the completion of the algebraic closure of the formal power seriesfield F p [[ t ]] with its natural absolute value, then char K = char k = p ; if K is thecomplex p − adic field C p , then k = F p , the algebraic closure of F p , and char K = 0 but char k = p .Given a ∈ K and r > , define D ( a, r ) := { z ∈ K : | z − a | K < r } and D ( a, r ) := { z ∈ K : | z − a | K ≤ r } . If r ∈ | K × | K , we say that D ( a, r ) is an open rational disk in K and D ( a, r ) is aclosed rational disk in K . If r
6∈ | K × | K , we call D ( a, r ) = D ( a, r ) an irrational disk.Let U ( a, r ) ⊂ K be a disk centered at a ∈ K with radius r > , that is, U has theform D ( a, r ) or D ( a, r ) . Then if b ∈ U ( a, r ) , we have U ( a, r ) = U ( b, r ) . Moreover,the radius r is the same as the diameter of U ( a, r ) , that is r = sup {| z − w | K : z, w ∈ U ( a, r ) } . Furthermore, if two disks have a nonempty intersection, then one mustcontain the other. Finally, we should mention here every disk in K is both openand closed under the topology of K .Let L := K [[ t Q ]] be the field of Hahn series over K . It consists of all formal sumsof the form P n ≥ a n t q n , where { q n } is an increasing sequence of rational numbersand a n ∈ K . Since Q is divisible under addition, the field L is algebraically closed.It can be equipped with a non-Archimedean absolute value | · | L by fixing a number ǫ ∈ (0 , and defining | P n ≥ a n t q n | L = ǫ n , where n is the smallest positiveinteger such that a n = 0 . With respect to | · | L , the field L is complete. Then thering of integer of the field L is O L = { z ∈ L : | z | L ≤ } = X n ≥ a n t q n : q n ≥ and the unique maximal ideal M L of O L consists of series with zero constant term,i.e. M L = { z ∈ L : | z | L < } = X n ≥ a n t q n : q n > . The residue field O L / M L is canonically isomorphic to K .2.2. The Berkovich projective line.
In this subsection, we summarize somefundamental properties of the Berkovich projective line, for details we refer [3,4,6].The Berkovich affine line A ( K ) is the set of all multiplicative seminorms onthe ring K [ z ] of polynomials over K , whose restriction to the field K ⊂ K [ z ] isequal to the given absolute value | · | K . For a ∈ K and r ≥ , let ξ a,r be theseminorm defined by | f | ξ a,r = sup z ∈ D ( a,r ) | f ( z ) | K . Then there are types of pointsin A ( K ) :1. Type I. ξ a, for some a ∈ K . HONGMING NIE
2. Type II. ξ a,r for some a ∈ K and r ∈ | K × | .3. Type III. ξ a,r for some a ∈ K and r / ∈ | K × | .4. Type IV. A limit of seminorms { ξ a i ,r i } i ≥ , where the corresponding sequenceof closed disks { D a i ,r i } i ≥ satisfies D a i +1 ,r i +1 ⊂ D a i ,r i and ∩ D a i ,r i = ∅ .We can identify K with the type I points in A ( K ) via a → ξ a, . The point ξ , ∈ A ( K ) is called the Gauss point and denoted by ξ G . We put the weaktopology on A ( K ) , which makes the map A ( K ) → [0 , + ∞ ) sending ξ to | f | ξ continuous for each f ∈ K [ z ] . Then A ( K ) is locally compact, Hausdorff anduniquely path-connected.The Berkovich projective line P ( K ) is obtained by gluing two copies of A ( K ) along A ( K ) \{ } via the map ξ → /ξ . Then we can associate the Gelfand topol-ogy on P ( K ) . The Berkovich projective line P ( K ) is a compact, Hausdorff,uniquely path-connected topological space and contains P ( K ) as a dense subset.The space P ( K ) has tree structure. For a point ξ ∈ P ( K ) , we can definean equivalence relation on P ( K ) \ { ξ } , that is, ξ ′ is equivalent to ξ ′′ if ξ ′ and ξ ′′ are in the same connected component of P ( K ) \ { ξ } . Such an equivalence class ~v is called a direction at ξ . We say that the set T ξ P ( K ) formed by all directions at ξ is the tangent space at ξ . For ~v ∈ T ξ P ( K ) , denote by B ξ ( ~v ) − the componentof P ( K ) \ { ξ } corresponding to the direction ~v . If ξ ∈ P ( K ) is a type I or IVpoint, T ξ P ( K ) consists of a single direction. If ξ ∈ P ( K ) is a type II point,the directions in T ξ P ( K ) are in one-to-one correspondence with the elements in P ( k ) . If ξ ∈ P ( K ) is a type III point, T ξ P ( K ) consists of two directions. Notethe Gauss point ξ G is a type II point. We can identify T ξ G P ( K ) to P ( k ) by thecorrespondence T ξ G P ( K ) → P ( k ) sending ~v x to x , where ~v x is the direction at ξ G such that B ξ G ( ~v x ) − contains all the type I points whose images are x under thecanonical reduction map P ( K ) → P ( k ) .2.3. Rational maps.
In this subsection, we consider rational maps over the field K and define an analytic family of rational maps over K . For rational maps over anon-Archimedean field, we refer [3–5].We first define analytic maps on a disk U ⊂ K . Definition 2.1.
Let U ⊂ K be a disk and z ∈ U . We say a map f : U → K isanalytic if f can be written as a power series f ( z ) = ∞ X n =0 c n ( z − z ) n ∈ K [[ z − z ]] , which converges for all z ∈ U . The smallest n such that c n = 0 is called the orderof f at z and denoted ord z ( f ) . It is easy to check that analytic property is independent of the choice of z ∈ U .Moreover, if U = D ( z , r ) is a rational closed disk, then P ∞ n =0 c n ( z − z ) n convergesfor each z ∈ U if and only if lim n →∞ | c n | K r n = 0 . For rational open or irrationaldisks, lim n →∞ | c n | K r n = 0 implies P ∞ n =0 c n ( z − z ) n converges, but the converse isnot true.We denote by P ( K ) := K ∪ {∞} the projective line over K . We define thespherical metric on P ( K ) as follows: for points z = [ x : y ] and w = [ u : v ] in P ( K ) , ∆( z, w ) := | xv − yu | K max {| x | K , | y | K } max {| u | K , | v | K } . ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 5
Equivalently, ∆( z, w ) := ( | z − w | K max { , | z | K } max { , | w | K } , if z, w ∈ K, { , | z | K } , if z ∈ K, w = ∞ . Recall a degree d ≥ rational map f : P ( K ) → P ( K ) is represented by a pair f , f ∈ K [ X, Y ] of degree d homogeneous polynomials with no common factors,that is, f ([ X : Y ]) = [ f ( X, Y ) : f ( X, Y )] for all [ X : Y ] ∈ P ( K ) . Equivalently,the map f can be considered as the quotient of two relatively prime polynomials,of which the greatest degree is d . Let Rat d ( K ) denote the set of rational maps ofdegree d over K . Then Rat d ( K ) can be naturally identified with an open subsetof P d +1 ( K ) via the map Rat d ( K ) → P d +1 ( K ) sending ( a d z d + · · · + a ) / ( b d z d + · · · + b ) to [ a d : · · · : a : b d : · · · : b ] .Let φ ( z ) ∈ K ( z ) be a rational map. Suppose z ∈ P ( K ) and set w = φ ( z ) .Pick ψ , ψ ∈ PGL ( K ) such that ψ (0) = z and ψ ( w ) = 0 , and define Φ = ψ ◦ φ ◦ ψ . The multiplicity m φ ( z ) of φ at z is the order of Φ at . The weight w φ ( z ) of φ at z is the order of Φ ′ at . If Φ ′ ( z ) ≡ , we set w φ ( z ) = ∞ . Thiscan happen: for example, if char K = p and φ ( z ) = z p , then φ ′ ( z ) = 0 for each z ∈ K . A point z ∈ P ( K ) is called a critical point of φ if w φ ( z ) > . Denote Crit( φ ) ⊂ P ( K ) for the set of all critical points of φ . If every point z ∈ P ( K ) isa critical point of φ , then we say φ is inseparable. Otherwise, φ is called separable.Recall that for every rational map φ ∈ K ( z ) , we can write φ ( z ) = φ ◦ φ ( z ) , where φ is a separable rational map and φ ( z ) = z p j for some j ≥ if the field K haspositive characteristic p > or φ ( z ) = z if the field K has characteristic zero.It is called the (in)separable decomposition of φ . The rational map φ is calledthe separable part of φ . We define the nontrivial degree deg φ := deg φ and thenontrivial critical set Crit ( φ ) := φ − (Crit( φ )) of φ . Definition 2.2.
Let U ⊂ K be a disk containing . A collection { f t } t ∈ U ⊂ P d +1 ( K ) is a -dimensional separable analytic family of degree d ≥ rationalmaps if the map F : U → P d +1 ( K ) sending t to f t is an analytic map such that f t ∈ Rat d ( K ) is separable for all t = 0 . If { f t } t ∈ U is a -dimensional separableanalytic family of degree rational maps, we call it a moving frame. Let U ⊂ K be a disk containing . We say { f t } t ∈ U ⊂ P d +1 ( K ) is a -dimensional analytic family of nontrivial degree d ′ ≥ rational maps if we canwrite f t = g t ◦ h , where { g t } t ∈ U ⊂ P d ′ +1 ( K ) is a -dimensional separable an-alytic family of degree d ′ ≥ rational maps and h ( z ) = z p j for some j ≥ if char K = p > or h ( z ) = z if char K = 0 . Remark 2.3. (1) We are really interested in the germ defined by an analyticfamily, so considering a small disk V ⊂ U containing if necessary, wecan always assume U = D (0 , r ) is a rational closed disk.(2) For an analytic family { f t } on U , we can write f t ( z ) = P t ( z ) /Q t ( z ) = ( a d ( t ) z d + · · · + a ( t )) / ( b d ( t ) z d + · · · + b ( t )) and denote by ℓ the minimum among the orders of the a i ( t ) and b j ( t ) , atthe origin, i, j = 1 , · · · , d . Let C = max ≤ i,j ≤ d { , | lim t → a i ( t ) /t ℓ | K , | lim t → b j ( t ) /t ℓ | K } HONGMING NIE
Let x ∈ K be an element such that | x | K = C . For t sufficiently small, byconsidering g t ( z ) = ( P t ( z ) /xt ℓ ) / ( Q t ( z ) /xt ℓ ) if necessary, we can assume { f t } ⊂ O K ( z ) and that f t has at least one coefficient with absolute value . Therefore, throughout this paper, for a rational map φ ( z ) ∈ K ( z ) , weassume φ ( z ) ∈ O K ( z ) and at least one coefficient has absolute value . For an analytic family (cid:26) f t ( z ) = a d ( t ) z d + · · · + a ( t ) b d ( t ) z d + · · · + b ( t ) (cid:27) ⊂ K ( z ) of degree d rational maps, let a d , · · · , a , b d , · · · , b be the power series expressionsof the coefficients a d ( t ) , · · · , a ( t ) , b d ( t ) , · · · , b ( t ) , respectively. Then the degree d rational map f : P ( L ) → P ( L ) given by f ( z ) = a d z d + · · · + a b d z d + · · · + b is called, following Kiwi, the rational map associated to { f t } . The rational map f induces a map from P ( L ) to itself. We use the same notation f for the inducedmap.2.4. Rescaling limits for an analytic family.
A non-Archimedean field is locallycompact if and only if it is discretely valued and has finite residue field [7]. Then K is not locally compact, hence neither is P ( K ) . Thus, we define the rescaling limitsfor an analytic family of rational maps over K in the following sense: Definition 2.4.
Let { f t } be an analytic family of rational maps of nontrivial degreeat least . A moving frame { M t } is called a rescaling for { f t } if there exist an integer q ≥ , a rational map g : P ( K ) → P ( K ) of nontrivial degree d ′ ≥ and a finitesubset S of P ( K ) such that, as t → , M − t ◦ f qt ◦ M t ( z ) → g ( z ) pointwise on P ( K ) \ S . We say g is a rescaling limit for { f t } in P ( K ) \ S . Theminimal q ≥ such that the above holds is called the period of the rescaling { M t } . Following Kiwi [18], we define the following equivalence relations on the set ofall rescalings.
Definition 2.5.
Two moving frames { M t } and { L t } are equivalent if there exists M ∈ Rat ( K ) such that M − t ◦ L t → M as t → . Definition 2.6.
Two rescalings { M t } and { L t } for an analytic family { f t } aredynamically dependent if there exist an integer l ≥ and a nonconstant rationalmap g such that L − t ◦ f l ◦ M t → g , as t → , pointwise outside some finite set. If { M t } and { L t } are two equivalent rescalings for an analytic family { f t } , thenthey are dynamically dependent. The converse is not true in general.3. Reductions
Recall K is any arbitrary complete algebraically closed non-Archimedean field.Let g ∈ O K ( z ) be a rational map. Then reducing the coefficients of g modulo M K and canceling common factors, we get a rational map ˜ g over the residue field k ,which is called the reduction of g . Now we can define a map ρ K : Rat d ( K ) → Rat ≤ d ( k ) , ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 7 where
Rat ≤ d ( k ) is the space of degree at most d rational maps over k , sending g toits reduction ˜ g . We call ρ K the reduction map for rational maps over K .We first state an easy proposition and omit the proof. Proposition 3.1.
Let φ ( z ) , ψ ( z ) ∈ K ( z ) be rational maps, and let ρ ( φ ) and ρ ( ψ ) be their reductions, respectively. Then(1) ρ ( φ · ψ ) = ρ ( φ ) · ρ ( ψ ) ,(2) ρ ( φ + ψ ) = ρ ( φ ) + ρ ( ψ ) ,(3) If deg ρ ( ψ ) ≥ , then ρ ( φ ◦ ψ ) = ρ ( φ ) ◦ ρ ( ψ ) . In Proposition 3.1 (3) , if deg ρ ( ψ ) = 0 , the situation is complicated. For example,let K be the completion of the formal Puiseux series over C and define rational maps φ ( z ) = z /t and ψ ( z ) = tz over K . Then ρ ( φ ◦ ψ )( z ) = z but ( ρ ( φ ) ◦ ρ ( ψ ))( z ) = ∞ since ρ ( φ ) = ∞ .Recall that L is the field of Hahn series over K . Since L is an algebraicallyclosed and complete non-Archimedean field, we can consider the reduction map ρ L : Rat d ( L ) → Rat ≤ d ( K ) of rational maps over L . Definition 3.2.
Let { f t } be an analytic family of degree d ≥ rational maps. Wesay { f t } has good reduction if the associated rational map f has good reduction, thatis, deg ρ L ( f ) = d . Otherwise, we say { f t } has bad reduction. If there is a movingframe { M t } ⊂ P ( K ) such that { M − t ◦ f t ◦ M t } has good reduction, we say that { f t } has potentially good reduction. Given f = [ f : f ] ∈ P d +1 ( K ) , we can write f = [ f : f ] = [ H f ˆ f : H f ˆ f ] = H f [ ˆ f : ˆ f ] = H f ˆ f , where H f = gcd( f , f ) is a homogeneous polynomial and ˆ f = [ ˆ f : ˆ f ] is a rationalmap of degree at most d . Proposition 3.3.
Suppose { f t } is an analytic family of degree d ≥ rational mapssuch that f t → H f ˆ f , as t → , in P d +1 ( K ) . Then, as t → , f t converges to ˆ f pointwise on P ( K ) \ { H f = 0 } .Proof. Write f t = [ P t : Q t ] and ˆ f = [ P : Q ] . Since f t converges to H f ˆ f in P d +1 ( K ) ,there is a λ ∈ K \ { } such that for any [ X : Y ] ∈ P ( K ) , as t → , P t ( X, Y ) converges to λH ( X, Y ) P ( X, Y ) and Q t ( X, Y ) converges to λH ( X, Y ) Q ( X, Y ) . Soif [ X : Y ]
6∈ { H f = 0 } , we have f t ([ X : Y ]) converges to [ P ( X, Y ) : Q ( X, Y )] .Hence f t converges to ˆ f pointwise on P ( K ) \ { H f = 0 } . (cid:3) Corollary 3.4.
Let { f t } be an analytic family of degree d ≥ rational maps. If deg ρ L ( f ) ≥ , then { M t = z } is a rescaling for { f t } with corresponding rescalinglimit ρ L ( f ) .Proof. Note as t → there is a homogeneous polynomial H ∈ K [ X, Y ] such that f t converges to Hρ L ( f ) in P d +1 ( K ) . The conclusion then follows Proposition 3.3. (cid:3) The converse of Proposition 3.3 is also true.
Proposition 3.5.
Let { f t } be an analytic family of degree d ≥ rational map andlet S ⊂ P ( K ) be a finite subset. Suppose f t converges to ˆ f pointwise, as t → , on P ( K ) \ S . Then there exists a homogeneous polynomial H of degree d − deg ˆ f withzeros in S such that f t → H ˆ f , as t → , in P d +1 ( K ) . HONGMING NIE
Proof.
Let f be the associated rational map of { f t } . Then there exists homogeneouspolynomial H such that f t ( z ) converges to Hρ L ( f ) , as t → , in P d +1 ( K ) . Thus, byProposition 3.3, ˆ f = ρ L ( f ) . It is easy to check H satisfies the required conditions. (cid:3) Corollary 3.6.
Suppose K has positive characteristic p > . Let { f t ( z ) } ⊂ K ( z ) be an analytic family of rational maps of nontrivial degree at least . Let f be theassociated rational map of { f t } . If f is inseparable, then all the rescaling limits of { f t } are inseparable.Proof. Let g be a rescaling limit for { f t } . Then by Definition 2.4 and Proposition3.5, there exist a rescaling { M t } , an integer q ≥ and homogeneous polynomial H such that M − t ◦ f qt ◦ M t → Hg . Note the associated rational map M − ◦ f q ◦ M of { M − t ◦ f qt ◦ M t } is inseparable since f is inseparable. Considering the coefficientsof g , we have the map g is inseparable. (cid:3) Berkovich Dynamics
In this section, we first summarize the properties of the dynamics on a Berkovichspace, see [3, 4, 14, 20]. Then we restate the results in [18], which are proven fora holomorphic family of rational maps over C . These results are still true for ananalytic family { f t ( z ) } of rational maps over a field K with characteristic zero.Recall that the Berkovich Julia set J Ber ( φ ) of a rational map φ : P ( L ) → P ( L ) is the set consisting of all points ξ ∈ P ( L ) such that ∪ n ≥ φ n ( U ) omitsfinitely many points of P ( L ) for any neighborhood U of ξ . The classical Juliaset J I ( φ ) is J Ber ( φ ) ∩ P ( L ) . Let ξ ∈ P ( L ) \ P ( L ) be a periodic point of φ ofperiod n ≥ . The multiplier λ of ξ is defined by the local degree of φ n at ξ , that is, λ := deg ξ ( φ n ) . If λ ≥ , we say ξ is repelling. If a periodic point ξ ∈ P ( L ) \ P ( L ) is repelling, then ξ is a type II point. Let O ⊂ P ( L ) be a n -cycle of φ . The basinof O is the interior of the set of points ξ ∈ P ( L ) such that, for all neighborhoods U of O , the orbit of ξ is eventually contained in U .Recall that the tangent space T ξ P ( L ) is the set of all directions at ξ ∈ P ( L ) .Let φ : P ( L ) → P ( L ) be a rational map. Then for any ~v ∈ T ξ P ( L ) , there is aunique ~w ∈ T φ ( ξ ) P ( L ) such that for any ξ ′ sufficiently near ξ , φ ( ξ ′ ) ∈ B φ ( x ) ( ~w ) − .Thus the rational map φ induces a map φ ∗ : T ξ P ( L ) → T φ ( ξ ) P ( L ) sending the direction ~v to the corresponding direction ~w . Proposition 4.1. [3, Corollary 9.25, Theorem 9.26, Corollary 9.27, Proposition9.41] Let φ : P ( L ) → P ( L ) be a rational map of degree at least . Then φ ( ξ G ) = ξ G if and only if deg ρ L ( φ ) ≥ . Moreover,(1) Assume φ ( ξ G ) = ξ G . Identifying T ξ G P ( L ) to P ( K ) , the following hold:(a) deg ξ G φ = deg ρ L ( φ ) ,(b) at the Gauss point ξ G , φ ∗ = ρ L ( φ ) on P ( K ) .(2) For ξ ∈ P ( L ) and ~v ∈ T ξ P ( L ) , the image φ ( B ξ ( ~v ) − ) always con-tains B φ ( ξ ) ( φ ∗ ~v ) − , and either φ ( B ξ ( ~v ) − ) = B φ ( ξ ) ( φ ∗ ~v ) − or φ ( B ξ ( ~v ) − ) = P ( L ) . There exists an integer m ≥ such that(a) if φ ( B ξ ( ~v ) − ) = B φ ( ξ ) ( φ ∗ ( ~v )) − , then each ζ ∈ B φ ( ξ ) ( φ ∗ ( ~v )) − has m preimages in B ξ ( ~v ) − , counting multiplicities; ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 9 (b) if φ ( B ξ ( ~v ) − ) = P ( L ) , there is an integer N ≥ m such that each ζ ∈ B φ ( ξ ) ( φ ∗ ( ~v )) − has N preimages in B ξ ( ~v ) − and each ζ ∈ P ( L ) \ B φ ( ξ ) ( φ ∗ ( ~v )) − has N − m preimages in B ξ ( ~v ) − , counting multiplicities. Based on Proposition 3.3 and Proposition 4.1, we have
Proposition 4.2. [18, Proposition 3.4, Lemma 3.6, Lemma 3.7] Let { f t ( z ) } ⊂ K ( z ) be an analytic family of rational maps of nontrivial degree at least , and let { M t } and { L t } be moving frames. Denote by f , M and L the associated rationalmaps. Then(1) For all l ≥ , the following are equivalent:(a) There exists a rational map g : P ( K ) → P ( K ) of degree at least d ≥ such that M − t ◦ f lt ◦ M t converges to g pointwise, as t → , on P ( K ) off a finite subset.(b) f l ( ξ ) = ξ , where ξ = M ( ξ G ) and deg ξ f l = d .In the case in which ( a ) and ( b ) hold, the map ( f l ) ∗ : T ξ P ( L ) → T ξ P ( L ) is conjugate via a M ∈ Rat ( K ) to g : P ( K ) → P ( K ) .(2) Moving frames { M t } and { L t } are equivalent if and only if M ( ξ G ) = L ( ξ G ) .(3) The following are equivalent:(a) f ◦ M ( ξ G ) = L ( ξ G ) .(b) As t → , L − t ◦ f t ◦ M t converges to some nonconstant rational map g : P ( K ) → P ( K ) pointwise outside some finite subset. Corollary 4.3.
Let { f t } ⊂ K ( z ) be an analytic family of degree at least rationalmaps. Suppose { f t } has (potentially) good reduction. Then there is at most onerescaling, up to equivalence, for { f t } , and this rescaling is of period .Proof. Let f be the associated rational map of { f t } . Then f has (potentially) goodreduction. Then the classical Julia set J I ( f ) = ∅ and the Berkovich Julia set J Ber ( f ) is a singleton set [3, Lemma 10.53]. Thus φ has no repelling periodic points of type Iand has only one repelling periodic point [3, Theorems 10.81,10.82]. By Proposition4.2, all the rescalings of { f t } are equivalent and they are of period . (cid:3) To relate the critical points of f and the rescaling limits of { f t } , we first statethe following non-Archimedean Rolle’s theorem: Lemma 4.4. [11, Application 1] Suppose K has characteristic zero and residuecharacteristic zero. Let φ ∈ K ( z ) be a rational map of degree at least . If φ hastwo distinct zeros in the closed disk D ( a, r ) , then it has a critical point in D ( a, r ) . We should mention here Lemma 4.4 is not true in general. If K has characteristiczero and residue characteristic p > , then under same assumptions, φ is onlyguaranteed to have a critical point in D ( a, r | p | − / ( p − K ) which is strictly larger than D ( a, r ) . If K has characteristic p > , consider the field L and φ ( z ) = z p − z ∈ L ( z ) .Then φ has only one critical point, which is ∞ ∈ P ( L ) . However, φ has p zerosin D (0 , ⊂ L . For more details about rational maps with one critical point, werefer [12].Applying the non-Archimedean Rolle’s theorem and using the same proof in [18],we have Proposition 4.5.
Suppose K has characteristic zero. Consider a rational map φ : P ( L ) → P ( L ) of degree at least . Let ξ ∈ P ( L ) be a type II point and let ~v ∈ T ξ P ( L ) . If φ is not injective on B ξ ( ~v ) − , then there is a critical point of φ in B ξ ( ~v ) − such that the corresponding critical value φ ( c ) ∈ B φ ( ξ ) ( φ ∗ ( ~v )) − . Proposition 4.6.
Suppose K has characteristic zero. Consider a rational map φ : P ( L ) → P ( L ) of degree at least . Let O be a type II periodic orbit ofperiod q ≥ of φ . Assume the basin of O is free of critical points of φ . Then, forall ξ ∈ O , every ~v ∈ T ξ P ( L ) with φ q ( B ξ ( ~v ) − ) = P ( L ) has a finite forwardorbit under ( φ q ) ∗ . Moreover, if deg( φ q ) ∗ ≥ , then ( φ q ) ∗ is postcritically finite. In the next section, we establish analogs of these two propositions in positivecharacteristic, and from this deduce our main result.5.
Rational maps over fields with positive characteristic
Assume that the field K has positive characteristic p > . A nonconstant rationalmap φ ∈ K ( z ) can then be written as φ ( z ) = φ ( z p j ) for some integer j ≥ , where φ is separable. Recall the ramification locus R φ = { ξ ∈ P ( K ) : deg ξ φ ≥ } and φ is tame if R φ contains finitely many points with valence at least . Wesay a rational map φ ∈ K ( z ) is tamely ramified if the characteristic of K doesnot divide the multiplicity m φ ( z ) for any z ∈ P ( K ) . The space P ( K ) \ P ( K ) carries a natural metrizable topology, the strong topology, see [3, 10]. With respectto this metric, there exists r > such that the ramification locus R φ is in an r -neighborhood of the connected hull Hull(Crit( φ )) of critical set if and only if φ is tamely ramified [11, Theorem E]. If φ is separable, the extreme case R φ ⊆ Hull(Crit( φ )) is equivalent to φ is tame [10, Corollary 7.13].We can prove the following non-Archimedean Rolle’s theorem for a separabletame rational map over a field K with positive characteristic. Lemma 5.1.
Suppose K has positive characteristic p > . Let φ ∈ K ( z ) be aseparable tame rational map of degree at least . If φ has two distinct zeros in theclosed disk D ( a, r ) , then it has a critical point in D ( a, r ) .Proof. Suppose there is no critical point in D ( a, r ) . Let ξ a,r ∈ P ( K ) be thepoint corresponding to the closed disk D ( a, r ) . Then ξ a,r Hull(Crit( φ )) . Let ~v ∈ T ξ a,r P ( K ) be the direction such that D ( a, r ) ⊂ P ( K ) \ B ξ a,r ( ~v ) − . Thenthe set P ( K ) \ B ξ a,r ( ~v ) − is disjoint with Hull(Crit( φ )) . So P ( K ) \ B ξ a,r ( ~v ) − ∩ R φ = ∅ . Since R φ is closed, there exist ξ ∈ P ( K ) and ~w ∈ T ξ P Ber ( K ) such that P ( K ) \ B ξ a,r ( ~v ) − ⊂ B ξ ( ~w ) − and B ξ ( ~w ) − ∩ R φ = ∅ . Hence φ is injective on B ξ ( ~w ) − [10,Corollary 3.8]. So φ is injective on P ( K ) \ B ξ a,r ( ~v ) − . Thus φ is injective on theclosed disk D ( a, r ) . So φ has at most one zero in D ( a, r ) . It is a contradiction. (cid:3) Applying Lemma 5.1 and the argument in [18, Lemma 4.2], we obtain an analogof Proposition 4.5:
Proposition 5.2.
Suppose K has positive characteristic p > and consider aseparable tame rational map φ : P ( L ) → P ( L ) of degree at least . Let ξ ∈ P ( L ) be a type II point and let ~v ∈ T ξ P ( L ) . If φ is not injective on B ξ ( ~v ) − ,then there is a critical point of φ in B ξ ( ~v ) − such that the corresponding criticalvalue φ ( c ) ∈ B φ ( ξ ) ( φ ∗ ( ~v )) − . We now prove an analogy of Proposition 4.6:
ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 11
Proposition 5.3.
Suppose K has positive characteristic p > . Consider a rationalmap φ : P ( L ) → P ( L ) of nontrivial degree at least and suppose the separablepart φ of φ is tame. Let O be a type II periodic orbit of period q ≥ of φ .Assume the basin of O is free of critical values of φ . Then, for all ξ ∈ O , every ~v ∈ T ξ P ( L ) with φ q ( B ξ ( ~v ) − ) = P ( L ) has a finite forward orbit under ( φ q ) ∗ .Proof. Let ~v ∈ T ξ P ( L ) such that φ q ( B ξ ( ~v ) − ) = P ( L ) and ~v has an infiniteforward orbit under ( φ q ) ∗ . We will show there exists a critical value of φ in thebasin of O . Let q ≥ be the smallest integer such that φ ( B φ ◦ φ q ( ξ ) (( φ ◦ φ q ) ∗ ( ~v )) − ) = P ( L ) . Then by Proposition 4.1 and Proposition 5.2, there is a critical point c ∈ Crit( φ ) such that φ ( c ) ∈ B φ q ( ξ ) (( φ q +1 ) ∗ ( ~v )) − . Now we show for each n ≥ q + 1 , B φ n ( ξ ) (( φ n ) ∗ ( ~v )) − contains a point in the forward orbit of a critical value of φ .By induction, suppose it holds for n = k ≥ q + 1 . If φ ( B φ k ( ξ ) (( φ k ) ∗ ( ~v )) − ) = B φ k +1 ( ξ ) (( φ k +1 ) ∗ ( ~v )) − , then B φ k +1 ( ξ ) (( φ k +1 ) ∗ ( ~v )) − contains a point in the for-ward orbit of a critical value of φ . If φ ( B φ k ( ξ ) (( φ k ) ∗ ( ~v )) − ) = P ( L ) , then φ ( B φ ◦ φ k ( ξ ) (( φ ◦ φ k ) ∗ ( ~v )) − ) = P ( L ) . By Proposition 4.1 and Proposition 5.2, B φ k +1 ( ξ ) (( φ k +1 ) ∗ ( ~v )) − contains a critical value of φ .Thus, for n large, B φ nq ( ξ ) (( φ nq ) ∗ ( ~v )) − contains a point in the forward orbit of acritical value of φ . Note for n sufficiently large, say n ≥ n , φ ( B φ nq ( ξ ) (( φ nq ) ∗ ( ~v )) − ) = P ( L ) . Suppose φ l ( φ ( c )) ∈ B φ n q ( ξ ) (( φ n q ) ∗ ( ~v )) − for some c ∈ Crit( φ ) and l ≥ , then φ nq + l ( φ ( c )) → ξ , as n → ∞ , in the weak topology. Thus φ ( c ) is in the basin ofthe periodic cycle O . (cid:3) Corollary 5.4.
Under the same assumptions in Proposition 5.3, if deg ( φ q ) ∗ ≥ ,then ( φ q ) ∗ is postcritically finite at the nontrivial critical points.Proof. Suppose ( φ q ) ∗ is not postcritically finite at the nontrivial critical points.Let ~v ∈ T ξ P ( L ) be a nontrivial critical point of ( φ q ) ∗ with infinite forwardorbit, then there exists j ≥ such that ( φ ◦ φ j ) − (Crit( φ )) ∩ B ξ ( ~v ) − = ∅ . If φ n ( B ξ ( ~v ) − ) = P ( L ) for all n ≥ , then φ (Crit( φ )) ∩ B φ j +1 ( ξ ) (( φ j +1 ) ∗ ( ~v )) − = ∅ . Hence for all n ≥ , φ n ( φ (Crit( φ ))) ∩ B φ n + j +1 ( ξ ) (( φ n + j +1 ) ∗ ( ~v )) − = ∅ . So there exists c ∈ Crit( φ ) such that φ ( c ) in the basin of O . If there exists n ≥ such that φ n ( B ξ ( ~v ) − ) = P ( L ) , then ~v has a finite forward orbit by Proposition5.3. (cid:3) Based on Proposition 4.6 and Corollary 5.4, applying the argument in [18], wecan prove Theorem 1.1.
Proof of Theorem 1.1.
Let { M (1) t } , · · · , { M ( n ) t } be pairwise dynamically indepen-dent rescalings for { f t } of periods q , · · · , q n such that the corresponding rescalinglimits are not postcritically finite at nontrivial critical points. Let f be the asso-ciated rational map of { f t } and M (1) , · · · , M ( n ) be the associated rational mapsof { M (1) t } , · · · , { M ( n ) t } . Let ξ j = M ( j ) ( ξ G ) ∈ P ( L ) for j = 1 , · · · , n . Then by Proposition 4.2, for all j = 1 , · · · , n , f q j ∗ : T ξ j P ( L ) → T ξ j P ( L ) is not postcriti-cally finite at the nontrivial critical points, and the points ξ , · · · , ξ n are in pairwisedistinct periodic orbits of f . Note the separable part of f has at most f − critical points, hence it has at most f − critical values. Then by Proposition4.6 for the case char K = 0 and Corollary 5.4 for the case char K > , we have n ≤ f − . (cid:3) Examples
In this section, we give some examples to illustrate rescaling limits in non-Archimedean dynamics. We refer [18] for more examples. All examples in [18]are holomorphic families over C , which can be considered as analytic families overnon-Archimedean fields.Now let p > be a prime number. Denote by K the field F p [[ s Q ]] of Hahnseries over F p with respect to its nontrivial non-Archimedean absolute value. Then char K = p > . Example 6.1.
Polynomials with quadratic separable parts.
Given sufficiently small t ∈ K \ { } . Consider the map f t ( z ) = ( z p − ( s + t ))( z p − s ) ∈ K ( z ) . Then associated map f of { f t } has separable part f ( z ) = ( z − ( s + t ))( z − s ) . Notethat f is tame if and only if p ≥ . In fact, if p = 2 , f has only one critical pointat ∞ .Note { f t } is an analytic family that has good reduction. Then, up to equivalence,the moving frame { M t ( z ) = z } is the unique possible rescaling. The correspondinglimit is g ( z ) = ( z p − s ) . If p = 2 , then g ( z ) = z + s has nontrivial degree . If p ≥ , g ( z ) = ( z p − s ) has separable part g ( z ) = ( z − s ) . Note Crit( g ) = { s, ∞} and g ( s ) has an infinite forward orbit under g . Thus g is a rescaling limit that isnot postcritically finite at nontrivial critical points. Example 6.2.
Connected Julia sets.
Let a ∈ K with < | a | K < and b ∈ K with | b | K = | b − | K = 1 . Define φ a,b ( z ) = az + 1 az + z ( z − z − b ) ∈ K ( z ) . Then the Berkovich Julia set J Ber ( φ a,b ) is connected but not contained in a linesegment [2].For sufficiently small t ∈ K \ { } , let a = t q , where q = 3 p + 1 , and fix b ∈ K .Define f t ( z ) = φ t q ,b ( z p ) . Then { f t } is a degenerated analytic family defined in aneighborhood of t = 0 . Let f be the associated map of { f t } . Then the separablepart f of f is f ( z ) = ( t q z + 1) / ( t q z + z ( z − z − b )) , which has a connectedBerkovich Julia set J Ber ( f ) .First the moving frame { M t ( z ) = z } is a rescaling of period with rescalinglimit g ( z ) = 1 z ( z − z − b ) ◦ z p . Note f maps the segment [ ξ , | t | q/ , ξ G ] isometrically onto the segment [ ξ , | t | − q/ , ξ G ] .And f maps the segment [ ξ G , ξ , | t | − q/ ] bijectively to the segment [ ξ G , ξ , | t | q/ ] andthe segment [ ξ , | t | − q/ , ξ , | t | − q/ ] bijectively to the segment [ ξ , | t | q/ , ξ G ] , stretching ESCALING LIMITS IN NON-ARCHIMEDEAN DYNAMICS 13 by a factor of , respectively. We may expect there exists a point ξ ,r ∈ P ( L ) such that f ( ξ ,r ) = ξ ,r . Indeed, we can choose r = | t | L . Let L t ( z ) = tz . Then themoving frame { L t ( z ) } is a rescaling of period leading to rescaling limit h ( z ) = b z ◦ z , if p = 2 , − b z ◦ z , if p = 3 , − b p z ◦ z p , if p ≥ . Example 6.3.
McMullen maps.
This example is an analog of [18, 2.4]. Given sufficiently small t ∈ K \ { } ,consider the map f t ( z ) = z p + t p z p ∈ K ( z ) . Then { f t } is a degenerate analytic family defined in a neighborhood of t = 0 . Theassociated map f of { f t } has separable part f ( z ) = z + t p /z . By [10, Corollary6.6], when p ≥ , the map f ( z ) is tame. In fact, when p = 3 , at every type IIpoint, φ has separable reduction. Then by [10, Corollary 7.13], φ is tame. When p = 2 , φ has a unique critical point. Thus, φ is tame if and only if p ≥ .Obviously, the moving frame { M t ( z ) = z } is a rescaling of period . The corre-sponding rescaling limit is g ( z ) = z p , which has a tame separable part g ( z ) = z .Note the nontrivial critical set Crit ( g ) = { , ∞} . So the rescaling limit g is post-critically finite at the nontrivial critical points.Moreover, the moving frame { L t ( z ) = tz } is a rescaling of period for { f t } ,which leads to the rescaling limit h ( z ) = z − p . The rescaling limit h ( z ) is alsopostcritically finite at the nontrivial critical points. Acknowledgements.
The author is grateful to the anonymous referee for valuablecomments. The author would like to thank Jan Kiwi for explanation of his workand useful suggestions. The author would also like to thank Robert Benedetto,Laura DeMarco and Xander Faber for useful comments.
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