aa r X i v : . [ m a t h . D S ] M a r Hyperbolic Equivariants of Rational Maps
Kenneth JacobsOctober 12, 2018
Abstract
Let K denote either R or C . In this article, we introduce two new equivariants associatedto a rational map f ∈ K ( z ). These objects naturally live on a real hyperbolic space, and carryinformation about the action of f on P ( K ). When K = C we relate the asymptotic behavior ofthese equivariants to the conformal barycenter of the measure of maximal entropy. We also givea complete description of these objects for rational maps of degree d = 1. The constructions inthis article are based on work of Rumely in the context of rational maps over non-Archimedeanfields; similarities between the two theories are highlighted throughout the article. In this paper we introduce two new equivariants and one new invariant associated to the conjugationof a rational map f ∈ K ( z ) by SL ( K ), where K = R or K = C . The first equivariant is a function R F defined on real hyperbolic [ K : R ] + 1 space that measures the distortion of the unit spherein K induced by homogeneous lifts F γ of SL ( K )-conjugates f γ . The second equivariant is theset of points at which R F is minimized. The motivation for these objects comes from analogousconstructions due to Rumely [18, 19] for rational maps defined over complete, algebraically closed,non-Archimedean fields; throughout the article we will explain the connections between ours andRumely’s constructions.Let K = R or C , and let f ∈ K ( z ) have degree d ≥
1. Fix a homogeneous lift F : K → K of f , given by a pair of coprime polynomials F = ( F , F ), F i ∈ K [ X, Y ] of degree d satisfying f ( z ) = F ( z, F ( z, . Let S K denote the unit sphere (with respect to the Euclidean norm || · || ) in K ,and let dσ K be the volume form on S K normalized to have total volume 1. The quantity R ( F ) = Z S K log || F z || dσ K is a measurement of distortion of the sphere induced by F ; this distortion can be expressed explicitlyin terms of Alexander’s projective capacity ([1]; see Theorem 3.4.1 below).For γ ∈ SL ( K ), let F γ ( z ) = γ − · F ( γ · z ), where γ acts on K by left multiplication. Thematrices that preserve the Euclidean norm on K form a group SU ( K ), and one checks directlythat R ( F τ ) = R ( F ) for τ ∈ SU ( K ). Thus, the function R F ( γ ) := R ( F γ ) defined on SL ( K )descends to a well defined function R F : SL ( K ) / SU ( K ) → R . We observe that the space SL ( K ) / SU ( K ) is naturally isometric to real hyperbolic [ K : R ] + 1space, which for the moment we denote h K . The following theorem collects the basic properties ofthis function: 1 heorem 1.0.1. Let f ∈ K ( z ) with degree d ≥ , and fix a homogeneous lift F of f . The function R F : h K → R is smooth, proper, and subharmonic with respect to the hyperbolic Laplacian. Inparticular, it attains a minimum on h K . Theorem 1.0.1 follows from explicit calculations in which we estimate the growth rate of R F as [ γ ] ‘approaches the boundary of h K ’ (Theorem 3.1.1) and provide an explicit expression for thehyperbolic Laplacian of R F (Theorem 3.2.1).The function R F is also equivariant, in the sense that R F ([ γ ]) = R F γ ([id]) where [id] is the(class of the) identity matrix in SL ( K ) / SU ( K ). Consequently, the minimal value of R F is aninvariant of the lift F , and we define Definition 1.0.1.
The min invariant of f ∈ K ( z ) is defined to be m K ( f ) := min [ γ ] ∈ SL ( K ) / SU ( K ) R F ([ γ ]) − d log | Res( F , F ) | ;here, Res( F , F ) is the homogeneous resultant of the lift F .The normalization via the resultant addresses the fact that R F scales logarithmically for differentlifts of F , i.e. if c ∈ K × then R cF = R F +log | c | . While there are many ways that one could normal-ize R F to address this issue of scaling, our choice is based on analogy with the non-Archimedeansetting where the analogous normalization via the resultant has arithmetic significance: in thatcontext, m K ( f ) = 0 if and only if f has potential good reduction (see [18]).When K = C , we will also analyze the setMin( f ) = { [ γ ] ∈ SL ( K ) / SU ( K ) : [ γ ] minimizes R F } . For this, the notion of conformal barycenters of measures on S will be useful. Douady andEarle introduced the notion of conformal barycenters in [11] in order to study conformally naturalextensions of homeomorphisms of the sphere to the hyperbolic ball. More precisely, the barycenterof an admissible measure µ on the sphere S is a distinguished point of the unit ball in R whoseexistence can be established by showing that it is the minimum of a certain convex function h µ .Details will be given below in Section 2.6.When K = C , let ω C denote the Fubini-Study form on P ( C ), and for f ∈ C ( z ) let ω f := f ∗ ω C + f ∗ ω C , where the pushforward is in the sense of measures. Then ω f is a positive measure on P ( C ) of totalmass d + 1; its pullback to S under stereographic projection will be denoted c ω f . The followingtheorem gives a geometric interpretation of the conjugates realizing the minimum of R F : Theorem 1.0.2.
Identifying SL ( C ) / SU ( C ) with the hyperbolic ball B ⊆ R , the hyperbolic gra-dient of R F is ∇ h R F ( ξ ) = Z S ζ d ω f γξ , where [ γ ξ ] ∈ SL ( C ) / SU ( C ) is the class corresponding to ξ ∈ B . In particular, if [ γ ] ∈ SL ( C ) / SU ( C ) is in Min( f ) , then Bary( d ω f γ ) is the origin in the unit ball. Thus, conjugates for which R F is minimized correspond to ‘balanced’ representatives of f , wherebalanced is understood in terms of the measure ω f . In the non-Archimedean setting, classes [ γ ]belonging to the analogue of Min( f ) correspond to conjugates f γ that have semi-stable reduction2n the sense of Geometric Invariant Theory ([19]). While there is no natural notion of reductionin the complex setting, one can view ‘minimizing R F ’ as a complex analogue of having semi-stablereduction, and Theorem 1.0.2 gives a geometric interpretation of what ‘semi-stable’ might meanin this context. It would be very interesting to understand whether the classes [ γ ] attaining theminimum have an interpretation in terms of the moduli space M d of degree d rational maps. Weremark that the use of barycenters in studying moduli-theoretic questions has already been carriedout by DeMarco [9], who has shown that the barycenter of the measure of maximal entropy can beused to give a compactification of M .We next turn our attention to the asymptotic behavior of R F . The following theorem describesthe limits of R F n and Min( f n ). Recall that h µ is a convex function on the unit ball (with thehyperbolic metric) introduced by Douady and Earle that is minimized precisely on the barycenterof measures on the sphere. Theorem 1.0.3.
Let f ∈ C ( z ) have degree d ≥ , and let µ f be its measure of maximal entropyon P ( C ) .1. The functions d n R F n , viewed as functions on the unit ball B ⊆ R , converge locally uniformlyto h µ f + C F for an explicit constant C F .2. The sets Min( f n ) , viewed as subsets of the unit ball B ⊆ R , converge in the Hausdorfftopology to the conformal barycenter Bary( µ f ) of the measure of maximal entropy. The investigations carried out in this article were motivated by an analogous investigation ofRumely [18, 19] in the context of non-Archimedean dynamics. When K is a complete, algebraicallyclosed, non-Archimedean valued field and f ∈ K ( z ), then the analogue of our function R F isRumely’s ord Res f (there is a canonical, arithmetically-motivated normalization of the lift F in thenon-Archimedean setting, see [18]); while we haven’t precisely defined R F in the non-Archimedeancontext, this can be done, and the resulting function satisfiesord Res f ( γ ) = − log | Res( F , F ) | + 2 dR F ( γ ) , where Res( F , F ) is the homogeneous resultant of the lift F . Thus, our Theorem 1.0.1 is re-capturesparts of [18] Theorem 1.1 in the complex setting. The non-Archimedean version of Theorem 1.0.3 iscontained in [15] Theorems 1 and 3. Throughout the paper, we will point out additional similaritiesbetween our R F and Rumely’s ord Res f . One part of the non-Archimedean picture that is lackingfrom our constructions is the crucial measure of a rational map (see [19], Definition 9 and Corollary6.5). It would be very interesting to develop an analogue of these measures in the Archimedeansetting.We remark that the non-Archimedean and Archimedean perspectives can be combined to givean invariant for rational maps defined over a global field k : by summing the min-invariant over allplaces of k one obtains a function ˆ h R : M d → R . It is natural to ask whether this is comparableto a Weil height. By work of Doyle-Jacobs-Rumely [12], the answer is yes when d = 2 and k isa function field; similar work of a VIGRE research group at the University of Georgia [14] givesan affirmative answer if one restricts to polynomials under affine conjugation, again defined overfunction fields. The general question is currently under investigation.The article is organized as follows: in Section 2 we provide the necessary background informa-tion for the remainder of the article. In Section 3 we establish basic properties of R F , includingexplicit growth formulas for R F at the boundary of h K and an explicit expression for its hyperbolicLaplacian. Section 4 discusses the asymptotic behavior of R F n and Min( f n ). Finally, in Section 5we compute Min( f ) and m K ( f ) explicitly for maps f of degree 1.3 cknowledgements The author would like to thank Robert Rumely and Laura DeMarco for fruitful discussions andencouragement in carrying out this project.
Let K denote either the field R or C . We endow K with the Euclidean norm, || ( X, Y ) || = p | X | + | Y | . Under the usual left action of SL ( K ) on K by left multiplication, the Euclideannorm is preserved by the subgroupSU ( K ) = ( SO(2) , K = R SU(2) , K = C . We endow P ( K ) with a volume form ω K as follows: first let dℓ K be given locally by dℓ K = ( dα , K = R i dα ∧ dα , K = C be the top form cooresponding to the usual Lebesgue measure on K . Then ω K is given locally by ω K := (cid:18)
11 + | α | (cid:19) [ K : R ] dℓ K π = ( α dαπ , K = R i π | α | ) dα ∧ dα , K = C . (1)Note that when K = C , this is simply the Fubini-Study form. In both cases, ω K has been normalizedso that R P ( K ) ω K = 1. Viewing SL ( K ) as acting on P ( K ) by fractional linear transformations, ω K is invariant under pullback by elements of SU ( K ).The chordal metric on P ( K ) is given || P, Q || = | P − Q | p | P | · p | Q | for P, Q ∈ K and || P, ∞|| = √ | P | . It is rotation invariant, in the sense that for γ ∈ SL ( K )viewed as acting by linear fractional transformations on P ( K ), we have || γ ( P ) , γ ( Q ) || = || P, Q || .Throughout this paper we consider rational maps f ∈ K ( z ) of degree d = deg f ≥
1. Ahomogeneous lift F of f is a polynomial endomorphism F : K → K given F = ( F , F ) forcoprime, homogeneous polynomials F i ∈ K [ X, Y ] satisfying f ( z ) = F ( z, F ( z, . When needed, we willwrite the F i with coefficients as F ( X, Y ) = a d X d + ... + a Y d F ( X, Y ) = b d X d + ... + b Y d with a i , b i ∈ K . The left multiplication of SL ( K ) on K induces a conjugation of F by F F γ : ( X, Y ) = γ − · F ( γ · ( X, Y ))4e write F γ = ( F γ , F γ ), where F γ , F γ are the component polynomials of F γ .We define R ( F ) = Z S K log || F ( X, Y ) || dσ K , where S K ⊆ K is the unit sphere and dσ K is the unit volume form on S K . This quantity canbe interpreted geometrically and will be explored further in Section 3.4; of greater interest for ourpurposes is the function on SL ( K ) given R F ( γ ) := R ( F γ ) . Note that the SU ( K )-invariance of σ K and S K implies that R F : SL ( K ) → R descends toa well-defined function R F : SL ( K ) / SU ( K ) → R . In the next section we will explain howSL ( K ) / SU ( K ) is isometric to a real hyperbolic space, and will utilize the geometry of this spaceto deduce properties of R F . It will also be useful to have an expression for R F as an integral on P ( K ): R ( F ) = Z P ( K ) log || F ( z, |||| ( z, || d ω K . (2)Let H K := { ( z, t ) : z ∈ K, t > } , and define P K ( z, t ) on H K by P K ( z, t ) = (cid:18) tt + | z | (cid:19) [ K : R ] . This function is the hyperbolic Poisson kernel, used to construct hyperbolic harmonic extensionsof functions g : P ( K ) → R (see [21], Section 5.6, and also Proposition 2.5.1 below). A directcalculation also shows that ( γ − z,t ) ∗ ω K ( α ) = P K ( α − z, t ) dℓ K ( α ) π (3)where γ z,t ( α ) = tα + z . Thus the change of variables formula gives Z P ( K ) Q ( γ z,t ( α )) ω K ( α ) = Z P ( K ) Q ( α ) ( γ z,t ) ∗ ω K ( α ) = Z P ( K ) Q ( w ) · P K ( z, t ; w ) dℓ K ( w )for any smooth function Q on P ( K ). In the different sections of the paper it will be convenient to work with different models of realhyperbolic 3 space. Let E K = R [ K : R ]+1 . We identify three models by specifying a space X , a metricd X , and a distinguished point in X which we denote (in all models) by j : • the upper half space model is the space ( H K , d H ) consisting of points ( z, t ) with z ∈ K and t >
0; the distinguished point is j = (0 , • the conformal ball model is the space ( B K , d B ), consisting of points ξ = rζ , where 0 ≤ r < ζ an element of the unit sphere in S K ⊆ E K ; the distinguished point is j = 0 is the zerovector in E K . • the quotient space (SL ( K ) / SU ( K ) , d SL ), consisting of SU ( K )-equivalence classes of ma-trices in SL ( K ); here, the distinguished point is j = [id], the equivalence class of the identitymatrix. 5he metrics d H and d B are the usual hyperbolic metrics on the respective model; see, e.g. [17]. Inparticular, if ( x, t ) , ( y, s ) ∈ H K , thencosh d H (( x, t ) , ( y, s )) = 1 + | x − y | + ( s − t ) st . (4)If x = y = 0, this reduces to give d H ((0 , t ) , (0 , s )) = (cid:12)(cid:12) log (cid:0) ts (cid:1)(cid:12)(cid:12) . The metric d SL on SL ( K ) / SU ( K )is perhaps less familiar, and will be defined below.We begin by recalling some standard decompositions of matrices γ ∈ SL ( K ). Lemma 2.2.1.
Let [ γ ] ∈ SL ( K ) / SU ( K ) .a) The class [ γ ] has a unique representative of the form γ = τ · η A , where τ ∈ SU ( K ) and η A = (cid:18) e A/ e − A/ (cid:19) for some A ≥ .b) The class [ γ ] has a unique representative of the form γ z,t = √ t z √ t √ t ! , (5) where z ∈ K and t > .Proof. a) By [2], Theorems 5.4 and 5.8 we can write γ = τ ητ ∗ where τ ∗ is the conjugate transposeof τ ∈ SU ( K ) and η is a real diagonal matrix; perhaps replacing τ by τ (cid:18) (cid:19) , we find that η = η A for A ≥ γ ′ ∈ SL ( K ), it amounts to solving γ ′ = γ z,t · τ for some z, t and τ ∈ SU ( K ). Uniqueness follows by showing that if γ z,t · γ − z ′ ,t ′ ∈ SU ( K ), then z = z ′ and t = t ′ .The quantity A appearing in part (a) of Lemma 2.2.1 can be used to define a metric onSL ( K ) / SU ( K ): Definition 2.2.1.
The metric d SL on SL ( K ) / SU ( K ) is defined d SL ([ γ ] , [ ω ]) = A , where τ · η A is the unique representative of [ ω − · γ ] appearing in Lemma 2.2.1 (a).We will show in Proposition 2.2.2 that this is indeed a metric on SL ( K ) / SU ( K ). Note thatthe following invariance property is already clear: for α, γ, ω ∈ SL ( K ), we have d SL ([ α · γ ] , [ α · ω ]) = d SL ([ γ ] , [ ω ]).We next recall an isometry between ( H K , d H ) and ( B K , d B ): Proposition 2.2.1.
The spaces ( H K , d H ) and ( B K , d B ) are isometric. Fix coordinates on theappropriate Euclidean space (depending on K ) so that H K = { ( z, t ) : z ∈ K, t > } and B K = { r · ζ : 0 ≤ r < , ζ ∈ S K } . Let σ K denote inversion in the sphere S K ((0 , , √ and let η K denote reflection in the hyperplane { ( x,
0) : x ∈ K } ⊆ E K . Then the map ι K : B K → H K given ι K = η K ◦ σ K defines an isometry between ( B K , d B ) and ( H K , d H ) . Moreover, this inversionextends to give stereographic projection Σ : S K → P ( K ) on the boundaries of B K and H K . It is perhaps easier to see this reduced formula using the metric tensor on H K given by ds H = ds K + dt t , where ds K is the Euclidean metric on K . roof. See, e.g, [17] Sections 4.5 and 4.6.There is a natural action of SL ( K ) on H K that preserves the metric d H , given by the actionof fractional linear transformations ([17], Theorem 4.6.2). For our purposes, the action for thematrices given in Lemma 2.2.1 are γ z,t · (0 ,
1) = ( z, t ) η A · (0 ,
1) = (0 , e A ) . By identifying H K to B K via the map ι K appearing in Proposition 2.2.1 we also obtain anaction of SL ( K ) on B K . There is also a natural action of SL ( K ) on SL ( K ) / SU ( K ) given byleft multiplication. In all cases, we find that SU ( K ) is the stabilizer subgroup of the distinguishedpoint j in each model.We now identify isometries between SL ( K ) / SU ( K ) and the other models of hyperbolic spacediscussed above: Proposition 2.2.2.
The maps Γ H : SL ( K ) / SU ( K ) → H K Γ B : SL ( K ) / SU ( K ) → B K defined by Γ H ([ γ ]) = γ · (0 , and Γ B ([ γ ]) = γ · are bijections, and the metric on SL ( K ) / SU ( K ) induced by either of these maps coincides with d SL introduced above.Proof. The fact that these maps are bijections follows from the fact that SU ( K ) is the stabilizersubgroup of the respective distinguished points j . To see that the metric induced by these mapsagrees with d SL , it’s enough to show that the metric induced by d H agrees with d SL (since theaction of SL ( K ) on B K was defined by the isometry ι K of H K and B K ). The metric d H is knownto be invariant under the left action of SL ( K ), and we also saw that d SL is invariant under theaction of SL ( K ) on SL ( K ) / SU ( K ). Therefore, we only need to check that d SL ([ γ ] , [id]) = d H ( γ · (0 , , (0 , γ ] ∈ SL ( K ) / SU ( K ).Let τ · η A be the unique representative of [ γ ] given in Lemma 2.2.1. Then d SL ([ γ ] , [id]) = A .We also compute d H ( γ · (0 , , (0 , d H (( τ · η A ) · (0 , , (0 , d H ( η A · (0 , , τ − · (0 , d H ( η A · (0 , , (0 , . Above we noted that η A sends (0 ,
1) to (0 , e A ), and from the formula for d H mentioned above wefind that d H ( γ. (0 , , (0 , d H ((0 , e A ) , (0 , e A ) − log(1) = A = d SL ([ γ ] , [id]) . We close this section with some remarks and conventions that will be used in the rest of thepaper: • If f : SL ( K ) / SU ( K ) → R is a function on SL ( K ) / SU ( K ), then we will often write f ( γ ( j )) = f ([ γ ]) for the corresponding function on H K . In a similar way, the function on B K will often be written f ( γ (0)) = f ([ γ ]). In all cases, the function name, f , will be the same,and we will rely on its argument to determine the domain.7 Functions and measures defined on P ( K ) or H K can be pulled back along stereographicprojection to be defined on S or B K ; we will denote these pullbacks by putting a hat b · overthe object. For example, if µ is a probability measure on P ( K ) then b µ = ( ι K ) ∗ µ , where ι K is stereographic projection (see Proposition 2.2.1). P ( C ) In the case K = C , we will make much use of techinques from potential theory. Here, we recall afew facts that we will need.Viewing points of P ( C ) in homogeneous coordinates [ x : x ], the patch U = { [ x : x ] : x =0 } will be given local coordinates z = x x and z . The operator dd c in these coordinates is definedon C functions by dd c ψ := iπ ψ zz dz ∧ dz . We will make use of the following well-known facts pertaining to dd c : Proposition 2.3.1.
Let ψ : C → R be C . Then1. For every f : C → C holomorphic, we have f ∗ dd c ψ = dd c ( ψ ◦ f ) .
2. For every C function φ : C → R , we have Z P ( C ) φ dd c ψ = Z P ( C ) ψ dd c φ . R F In this section, we collect several for R F that will be useful throughout the paper. The firstcomputes R F ([id]) = R ( F ) when F is a linear map: Lemma 2.4.1. If M ∈ SL ( K ) with M = τ · η A · σ for τ, σ ∈ SU ( K ) and η A as in Lemma 2.2.1(a),then Z S K log || M · ( X, Y ) ⊤ || dσ K = log(1 + e A ) − A − log 2 , K = R − + A (cid:16) e A +1 e A − (cid:17) , K = C . Proof.
Note that the SU ( K )- invariance of S K and dσ K allows us to reduce our calculations to Z S K log || M · ( X, Y ) ⊤ || dσ K = Z S K log || η A · ( X, Y ) ⊤ || dσ K = 12 Z S K log (cid:0) e A | X | + e − A | Y | (cid:1) dσ K . Expressing this as an integral over P ( K ) gives Z S K log || M · ( X, Y ) ⊤ || dσ K = 12 Z P ( K ) log (cid:18) e A | α | + 1 | α | + 1 (cid:19) ω K ( α ) − A
2= 12 Z P ( K ) log( e A | α | + 1) ω K ( α ) − Z P ( K ) log( | α | + 1) ω K ( α ) − A . An explicit – but tedious – calculation of these integrals in local coordinates gives the expressionsasserted in the statement of the lemma. 8e next give a slight refinement of the affine expression for R F given in (2): Proposition 2.4.1.
Let f ∈ K ( z ) and γ ∈ SL ( K ) be represented by γ z,t as in Lemma 2.2.1. Then R F ([ γ ]) = 12 Z P ( K ) log | f ( w ) | · (cid:0) | f ( w ) − z | + t (cid:1) t d +1 P K ( α − z, t ) dℓ K π + c K , where c K = − d R P ( K ) log(1 + | α | ) ω K ( α ) = (cid:26) − d log 2 , K = R − d , K = C .Proof. Putting the integrand of R F into affine coordinates yieldslog || F γ ( X, Y ) |||| X, Y || d = 12 log t − ( d +1) (cid:16) | F ( tα + z, − zF ( tα + z, | + t | F ( tα + z, | (cid:17) ( | α | + 1) d = 12 log | f ( γ ( α )) | · (cid:0) | f ( γ ( α )) − z | + t (cid:1) t d +1 − d | α | ) . (6)Integrating the second term against ω K gives, after a direct calculation, the constant c K . Integratingthe first term against ω K and using (3) then gives the asserted expression.When K = C , we can use the complex structure to deduce the following useful expression for R F : Proposition 2.4.2.
Let f ∈ C ( z ) , and let [ γ ] ∈ SL ( C ) / SU ( C ) be represented by γ z,t as inLemma 2.2.1. Define ψ ( α ; z, t ) = − log || γ − z,t ( α ) , ∞|| and ω z,t = ( γ − z,t ) ∗ ω C . R F ([ γ ]) = d −
12 log t + Z P ( C ) ψ ( α ; z, t ) f ∗ ω z,t − d c ( z, t ; f ) , (7) where c ( z, t ; f ) = | a d − zb d | + t | b d | .Proof. Define ψ ( z ) = log(1+ | z | ); an explicit calculation shows that dd c ψ = ω C − δ ∞ as currentson P ( C ). We let [ γ ] ∈ SL ( C ) / SU ( C ) be represented by γ z,t as in Lemma 2.2.1; as a fractionallinear transformation on C , this can be realized as γ z,t ( α ) = tα + z . Note that ψ ( α ; z, t ) = ( γ − z,t ) ∗ ψ ( α ) , so that by Proposition 2.3.1 we have dd c ψ ( · ; z, t ) = ( γ z,t − ) ∗ ω C − δ ∞ = ω z,t − δ ∞ . (8)In local coordinates, ω z,t = i π ∂ α ∂ α ψ ( α ; z, t ) dα ∧ dα .From (6) we find thatlog || F γ ( X, Y ) |||| X, Y || d = − ( d + 1)2 log t + 12 log | f ( γ ( α )) − zf ( γ ( α )) | + t | f ( γ ( α ) | ( | α | + 1) d , (9)where α = XY is a local coordinate on P ( C ). As | α | → ∞ , the second term tends to d log t + log c ( z, t ; f ), where c ( z, t ; f ) is given by c ( z, t ; f ) := (cid:26) | a d | , f ( ∞ ) = ∞ ( | f ( ∞ ) − z | + t ) | b d | , f ( ∞ ) = ∞ = | a d − zb d | + t | b d | . R F , we find that R F ( γ ) = − d + 12 log t + 12 Z P ( C ) log | f ( γ ( α )) − zf ( γ ( α )) | + t | f ( γ ( α ) | ( | α | + 1) d ω C = − d + 12 log t + 12 Z P ( C ) log | f ( γ ( α )) − zf ( γ ( α )) | + t | f ( γ ( α ) | ( | α | + 1) d ( ω C − δ ∞ ) + 12 log c ( z, t ; f ) + d log t = d −
12 log t + 12 Z P ( C ) log | f ( α ) − zf ( α ) | + t | f ( α ) | ( | γ − ( α ) | + 1) d ( γ − ) ∗ dd c ψ + 12 log c ( z, t ; f )= d −
12 log t + Z P ( C ) ( f ∗ ψ ( α ; z, t ) − dψ ( α ; z, t ) + log | f ( α ) | ) dd c ψ ( · ; z, t ) + 12 log c ( z, t ; f ) . (10)We apply Proposition 2.3.1 to pass the dd c onto the integrand; note that dd c f ∗ ψ ( · ; z, t ) = f ∗ ( ω z,t − δ ∞ ) = f ∗ ω z,t − X f ( p i )= ∞ ,p i = ∞ δ p i + ( d − deg( f )) δ ∞ ,dd c log | f ( α ) | = X f ( p i )= ∞ , p i = ∞ δ p i − deg( f ) δ ∞ . Inserting these into (10) and simplifying yields R F ( γ ) = d −
12 log t + Z P ( C ) ψ ( α ; z, t ) ( f ∗ ω z,t − d · ω z,t ) + 12 log c ( z, t ; f )= d −
12 log t + Z P ( C ) ψ ( α ; z, t ) f ∗ ω z,t − d Z P ( C ) ( γ − ) ∗ ψ ( α )( γ − ) ∗ ω C + 12 log c ( z, t ; f )= d −
12 log t + Z P ( C ) ψ ( α ; z, t ) f ∗ ω z,t − d c ( z, t ; f ) , which is the asserted formula. The various hyperbolic spaces introduced in Section 2.2 each come equipped with a Laplace op-erator; in the geometric setting (i.e. H K and B K ) this is the Laplace-Beltrami operator that canbe computed in terms of the metric tensor for d H and d B respectively. On SL ( K ) / SU ( K ), theLaplace-Beltrami operator can be interpreted in terms of the Casimir element of the universal en-veloping algebra of sl ( K ), though we will not make use of this perspective in the present article(see [13]).Viewing H K ⊆ E K with coordinates ( z, t ) where z ∈ K, t >
0, we have (e.g. [21], Exercise3.5.11) ∆ H K h = (cid:26) t ∆ std , K = R t ∆ std − t ∂∂t , K = C , where ∆ std is the standard Laplace operator on E K given ∆ std = P [ K : R ]+1 i =1 ∂ ∂x i .On B K , we will use an expression for the hyperbolic Laplacian in terms of spherical (or polar)coordinates on E K (see, e.g. [21], Exercise 3.5.6)∆ B K h = 1 − r r (cid:0) (1 − r ) N + ([ K : R ] − r ) N + (1 − r )∆ σ (cid:1) , N = r ∂∂r and ∆ σ is the part of ∆ std corresponding to the angular coordinates.A continuous function g : P ( K ) → R can be extended to a function H { g } : H K → R via theformula H { g } (( z, t )) := ( g ∗ P K )( z, t ) = Z K g ( α ) P K ( α − z, t ) dℓ K , that is hyperbolic harmonic, i.e ∆ H K h H { g } = 0. The function H { g } extends g in the sense that, if z ∈ P ( K ) is viewed as a point in the ideal boundary of H K , then lim H K ∋ w → z H { g } ( w ) = g ( z ) (see[21], Theorem 5.6.2). The next result expresses this extension as a function on SL ( K ) / SU ( K ): Proposition 2.5.1.
Let g : P ( K ) → R be a continuous function. Given [ γ ] ∈ SL ( K ) / SU ( K ) ,let ˇ g ([ γ ]) := Z P ( K ) g ( γ ( α )) ω K . Then H { g } ( γ ( j )) = ˇ g ([ γ ]) . Proof.
By Lemma 2.2.1 each [ γ ] ∈ SL ( K ) / SU ( K ) can be represented by a matrix γ = γ z,t = (cid:18) √ t z/ √ t / √ t (cid:19) , where z ∈ K and t >
0. This, in turn, corresponds to an affine map γ z,t ( α ) = tα + z acting on P ( K ). By (3), ( γ − z,t ) ∗ ω K = P K ( α − z, t ) dℓ K π . Therefore, ˇ g ([ γ z,t ]) = Z P ( K ) g ( γ z,t ( α )) ω K = Z P ( K ) g ( α )( γ − z,t ) ∗ ω K = Z K g ( α ) P K ( α − z, t ) dℓ K = ( g ∗ P K )( z, t ) = H { g } ( γ z,t ( j ))as asserted.As an application of this – and because we will need it later – we have Proposition 2.5.2.
Let ν be a probability measure on S = ∂ B C so that the pushforward Σ ∗ ν by stereographic projection has continuous potentials on P ( C ) , i.e. there is a continuous function g ν : P ( C ) → R so that dd c g ν = Σ ∗ ν − δ ∞ as measures on P ( C ) .For any point ξ ∈ S and any sequence of points ξ n ∈ B C with ξ n → ξ , we have Z S log | ζ − ξ n | dν ( ζ ) → Z S log | ζ − ξ | dν ( ζ ) . Proof.
By rotating the sphere, we can assume that ξ = (0 , , −
1) is the ‘south pole’; notice thatΣ( ξ ) = 0 ∈ C . We need the following lemma: 11 emma 2.5.1. Let ζ, ξ ∈ B C correspond to points ( z, t ) , ( z ′ , t ′ ) ∈ H C (resp.), and write ~n = (0 , , .Then | ζ − ξ | = ( | z − z ′ | +( t ′ − t ) ) ( | z | +( t +1) ) · ( | z ′ | +( t ′ +1) ) , ζ, ξ = ~n | z ′ | +( t ′ +1) , ζ = (0 , , , ξ = ~n | z | +( t +1) , ζ = (0 , , , ξ = ~n , ζ = ξ = ~n where | · | on the left side is chordal distance, and on the right side is used to denote the usualabsolute value on C .Proof. Note that the last three cases can be obtained from the first by taking limits as ζ → ~n and/ or ξ → ~n . So, it suffices to prove the first formula.The map ι C appearing in Proposition 2.2.1 can be computed explicitly as ι C = σ ◦ η , where σ is inversion in the sphere centered at ~n of radius √
2, and η is reflection in the hyperplane { ( x, y,
0) : x, y ∈ R } ⊆ E C . Let p = η ( z, t ) = ( z, − t ) and q = η ( z ′ , t ′ ) = ( z ′ , − t ′ ). Then by [17]Theorem 4.1.3 we have | ζ − ξ | = | σ ( p ) − σ ( q ) | = 2 | p − q || p − ~n | · | q − ~n | , where here all of the absolute values are in terms of the Euclidean metric on E C . Inserting thedefinition of p, q yields | ζ − ξ | = 4( | z − z ′ | + ( t ′ − t ) ( | z | + ( t + 1) ) · ( | z ′ | + ( t ′ + 1) ) , where now on the right side we are using the absolute values to denote the usual distance on C .We now return to the proof of Proposition 2.5.2. Note that in our case, for ζ, ξ, ξ ∈ B C corresponding to ( z, , ( z ′ , t ′ ) , (0 , ∈ H C (resp.), we find | ζ − ξ | = 4( | z − z ′ | + ( t ′ ) )( | z | + 1) · ( | z ′ | + ( t ′ + 1) ) | ζ − ξ | = 4 | z | | z | + 1 | ξ − ξ | = 4( | z ′ | + ( t ′ ) ) | z ′ | + ( t ′ + 1) . In particular, | ζ − ξ | | ζ − ξ | = | z − z ′ | + ( t ′ ) | z | · ( | z ′ | + ( t ′ + 1) ) . (11)Taking logarithms, integrating, and doing some simplification yields Z S log | ζ − ξ || ζ − ξ | dν ( ζ ) = 12 Z P ( C ) log | z − z ′ | + ( t ′ ) | z | d Σ ∗ ν ( z ) − log( | z ′ | + ( t ′ + 1) ) . Note that − log( | z ′ | + ( t ′ + 1) ) → ξ → ξ , since in this case ( z ′ , t ′ ) → (0 , ξ → ξ . Note that the integrand tendsto 0 as | z | → ∞ , so that12 Z P ( C ) log | z − z ′ | + ( t ′ ) | z | d Σ ∗ ν ( z ) = 12 Z P ( C ) log | z − z ′ | + ( t ′ ) | z | dd c g ν ( z ) , g ν is the potential function referred to in the statement of the proposition. Applying inte-gration by parts to transfer the dd c to the integrand, we have12 Z P ( C ) log | z − z ′ | + ( t ′ ) | z | dd c g ν = 12 Z P ( C ) g ν dd c (cid:18) log | z − z ′ | + ( t ′ ) | z | (cid:19) . (12)The dd c here can be computed explicitly. Letting γ z ′ ,t ′ = (cid:18) √ t ′ z ′ / √ t ′ / √ t ′ (cid:19) , by (8) we find dd c (cid:18) log | z − z ′ | + ( t ′ ) | z | (cid:19) = 2( γ − z ′ ,t ′ ) ∗ ω C − δ . Inserting this into (12) and applying Proposition 2.5.1 yields12 Z P ( C ) g ν dd c (cid:18) log | z − z ′ | + ( t ′ ) | z | (cid:19) = Z P ( C ) g ν ◦ γ z ′ ,t ′ − g ν (0)= H { g ν } ( z ′ , t ′ ) − g ν (0) . Since H { g ν } extends continuously to H C , giving g ν on the boundary P ( C ), we see that as ξ → ξ ,H { g ν } ( z ′ , t ′ ) − g ν (0) →
0, which finishes the proposition.
In the case that K = C , we will make use of the conformal barycenter of an admissible measure on S , an idea originally due to Douady and Earle [11].A probability measure µ on S will be called admissible if no point z ∈ S has mass ≥ .In this case, the function h µ ( z ) = − R S log (cid:16) −| z | | z − ζ | (cid:17) dµ ( ζ ) on B C is convex with respect to thehyperbolic metric and attains a unique minimum at a point Bary( µ ) of B C . The barycenter satisfies γ (Bary( µ )) = Bary( γ ∗ µ ) for all orientation-preserving automorphisms of P ( C ) ≃ S . We need thefollowing Proposition: Proposition 2.6.1.
Let µ be a probability measure on S , and let h µ ( z ) := − R S log (cid:16) −| z | | z − ζ | (cid:17) dµ ( ζ ) be the Douady-Earle function on B C . Then ∆ B C h h µ = 4 . Proof.
Endow B with the usual Euclidean coordinates ( x , x , x ). The hyperbolic Laplacian on B can be written as ∆ B C h = (1 − r ) ∆ std + 2(1 − r ) X i =1 x i ∂∂x i , where ∆ std = P i =1 ∂ ∂x i is the standard Euclidean Laplacian on R . By direct computation we findthat ∆ B C h (log(1 − | z | )) = − − | z | , and for fixed ζ ∈ S we find ∆ B C h (log | z − ζ | ) = 2(1 − | z | ) . The above computations imply that∆ B C h ( h µ ) = − (cid:18) ( − − r ) − Z (1 − r ) dµ ( ζ ) (cid:19) = 4 . Basic Properties of R F R F is Proper Throughout this section, we will work with the quotient space model of hyperbolic space (SL ( K ) / SU ( K ) , d SL ). Theorem 3.1.1.
Fix a homogeneous lift F of f . The map R F is smooth and proper; in particular,there are constants C ( F ) , C ( F ) so that for any [ γ ] ∈ SL ( K ) we have d − d SL ([ γ ] , [ id ]) + log C ( F ) − d ≤ R F ([ γ ]) ≤ d + 12 d SL ([ γ ] , [ id ]) + log C ( F ) . Lemma 3.1.1.
Let η A = (cid:18) e A/ e − A/ (cid:19) ∈ SL ( K ) . Then for any ( X, Y ) ∈ K , we find e −| A | / || X, Y || ≤ || η A ( X, Y ) || ≤ e | A | / || X, Y || . Proof.
Note that || η A ( X, Y ) || = e A | X | + e − A | Y | ≤ max( e A , e − A )( | X | + | Y | ) ;taking square roots establishes the upper bound. The lower bound follows by applying the upperbound to ( ˜ X, ˜ Y ) = η A ( X, Y ). Lemma 3.1.2.
Let F = [ F , F ] be a polynomial endomorphism of K with F, G homogeneous andcoprime of degree d ≥ . There exist constants C ( F ) , C ( F ) such that, for any X, Y ∈ K , C ( F ) · || X, Y || d ≤ || F ( X, Y ) || ≤ C ( F ) · || X, Y || d . (13) Moreover, the constants C and C are SU ( K ) -invariant in the sense that C ( F τ ) = C ( F ) and C ( F τ ) = C ( F ) for any τ ∈ SU ( K ) .Proof. For
X, Y ∈ K with || X, Y || = 1, we havemin || ˜ X, ˜ Y || =1 || F ( ˜ X , ˜ Y ) || ≤ || F ( X, Y ) || ≤ max || ˜ X, ˜ Y || =1 || F ( ˜ X , ˜ Y ) || The general case reduces to this by dividing the expression in (13) by || X, Y || d and using the ho-mogeneity of F . We take C ( F ) := min || ˜ X, ˜ Y || =1 || F ( ˜ X, ˜ Y ) || and C ( F ) := max || ˜ X, ˜ Y || =1 || F ( ˜ X , ˜ Y ) || ,which we observe are SU ( K ) invariant because SU ( K ) preserves the norm ||· , ·|| . Proof of Theorem 3.1.1.
The smoothness of R F can be seen directly from its definition: R F ([ γ ]) = Z S log || F γ ( X, Y ) || d Vol S . For the order estimates, recall that in local coordinates on P R F can be expressed as an integralover P ( K ) as R F ( γ ) = Z P ( K ) log || F γ ( X, Y ) |||| X, Y || d ω K . Write γ = τ · η A with A >
0, and let Ψ := F τ so that R F ( γ ) = Z P ( K ) log || Ψ η A ( X, Y ) |||| X, Y || d ω K . − A Z P ( K ) log || Ψ( η A ( X, Y )) |||| X, Y || d ω K ≤ R F ( γ ) ≤ A Z P ( K ) log || Ψ( η A ( X, Y )) |||| X, Y || d ω K . (14)We now apply Lemma 3.1.2 to estimate the integral I := R P ( K ) log || Ψ( η A ( X,Y )) |||| X,Y || d ω K aslog C ( F ) + d Z P ( K ) log || η A ( X, Y ) |||| X, Y || d ω K ≤ I ≤ log C ( F ) + d Z P ( K ) log || η A ( X, Y ) |||| X, Y || d ω K . (15)Let I := d Z P ( K ) log || η A ( X, Y ) |||| X, Y || ω K . In Lemma 2.4.1 the integral in I was computed explicitly as12 Z P ( K ) log || η A · ( X, Y ) |||| X, Y || ω K = ( log(1 + e A ) − A − log 2 , K = R − + A (cid:16) e A +1 e A − (cid:17) , K = C . When K = R , note that A ≤ log(1 + e A ) ≤ A + log 2 , whereby we obtain ( A − d ≤ I ≤ A d . (16)When K = C , note that for A > A ≤ A ( e A + 1) e A − ≤ A + 1From this, we find that ( A − d ≤ I ≤ A d . (17)Combining the estimates in (14), (15), (17), and (16), we obtain the asserted bounds for A > d − A + log C ( F ) − d ≤ R F ( γ ) ≤ d + 12 A + log C ( F ) . Recalling that A = d SL ([ γ ] , [id]), we see that we are done. Corollary 3.1.1. R F attains a minimum.Proof. Let M = R F ([id]), and set e = d − ( M − log C ( F )). Theorem 3.1.1 implies that for d SL ([ γ ] , [id]) > e we have R F ([ γ ]) > M (note that the constant C ( F ) is SU ( K ) invariant). Inparticular, R F must attain a minimum on the compact set B d SL ([id] , e ), which is necessarily lessthan or equal to M ; off of this compact set, we know that R F is strictly larger than M . Thus, theminimum attained on B d SL ([id] , e ) is a global minimum. Definition 3.1.1.
Let f ∈ K ( z ), and let F be a homogeneous lift of f . Then the min-invariant of f is the quantity m K ( f ) = min [ γ ] ∈ SL ( K ) / SU ( K ) R F ([ γ ]) − d log | Res( F , F ) | .
15t is expected that the min-invariant is related to the multipliers of the fixed points of f . Thiscan be shown explicitly in the case that d = 1; see Section 5 below. In the non-Archimedean setting,the connection between the minimal value of ord Res f and the multipliers of f is also known tohold for quadratic rational maps [12] and cubic polynomials [14].Also in the non-Archimedean setting, the function ord Res f – which we recall is the analogueof R F in that context – is minimized on what Rumely calls the ‘minimal resultant locus’, denotedMinResLoc( f ). Rumely shows that this set is either a single point or a segment ([18] Theorem 1.1),and that conjugates attaining this minimum correspond to maps with semistable reduction (in thesense of GIT).The following Theorem shows that the sublevel sets of R F are bounded, and will be used inSection 4.2 below (compare with [18] Theorem 1.1): Corollary 3.1.2.
Suppose that the degree of f is d ≥ . The functions R F n are level bounded, i.e.for any α ∈ R , there is an R > depending only on α and F such that { [ γ ] ∈ SL ( K ) / SU ( K ) : R F n ([ γ ]) ≤ α } ⊆ B R ([ id ]) for all n . Here, B ǫ ([ γ ]) is the ǫ -ball around [ γ ] in SL ( K ) / SU ( K ) with respect to the metric d SL .Proof. The lower bound in Theorem 3.1.1 for the family is d n − d SL ([ γ ] , [id]) + log C ( F n ) ≤ R F n ([ γ ]) , (18)where C ( F n ) = min || ˜ X, ˜ Y || =1 || F n ( ˜ X, ˜ Y ) || . We first observe that that C ( F n ) ≥ C ( F ) dn − d − , which follows inductively from the fact that || F n ( X, Y ) || = || F ( F n − ( X, Y )) || ≥ C ( F ) · || F n − ( X, Y ) || d . Thus (18) becomes d n − d SL ([ γ ] , [id]) + d n − d − C ( F ) ≤ R F n ([ γ ]) . Now let α ∈ R . If [ γ ] ∈ SL ( K ) / SU ( K ) is chosen so that R F n ([ γ ]) ≤ α , then d SL ([ γ ] , [id]) ≤ d n − α − d −
1) log C ( F ) ≤ , α ) − d −
1) log C ( F ) . Setting R = 2 max(0 , α ) − d − log C ( F ), the lemma is proved. R F is subharmonic In this section, it will be most convenient to work with the upper half-space model ( H K , d H ) ofhyperbolic space. We will show 16 heorem 3.2.1. For any [ γ ] ∈ SL ( K ) / SU ( K ) , we have ∆ H K h R F ([ γ ]) = [ K : R ] d − Z P ( K ) || f γ ( w ) , w || ω K ! , where ∆ h is the Laplace-Beltrami operator for ( H K , d H ) . In particular, R F is strictly subharmonicfor F not equal to the identity map. The proof rests on several explicit calculations. Recall first that we showed in Proposition 2.4.1that R F ([ γ ]) = 12 Z P ( K ) log | f ( w ) | · (cid:0) | f ( w ) − z | + t (cid:1) t d +1 P K ( α − z, t ) dℓ K π + c K (19)for an explicit constant c K . Let L ( z, t ; w ) = log( | f ( w ) − z | + t ) + 2 log | f ( w ) | − ( d + 1) log t be the logarithmic component of the integrand above. We begin by computing its hyperbolicLaplacian: Lemma 3.2.1.
Fix w ∈ K . Let z = z + iz be a complex number, and write f ( w ) = f Re + if Im .If K = R , we assume z = z , f ( w ) = f Re . ∂∂t L ( z, t ; w ) = 2 t | f ( w ) − z | + t − d + 1 t∂∂z L ( z, t ; w ) = − f Re − z ) | f ( w ) − z | + t ∂∂z L ( z, t ; w ) = − f Im − z ) | f ( w ) − z | + t ∂ ∂t L ( z, t ; w ) = 2 | f ( w ) − z | − t ( | f ( w ) − z | + t ) + d + 1 t ∂ ∂z L ( z, t ; w ) = 2( f Im − z ) − f Re − z ) + 2 t ( | f ( w ) − z | + t ) ∂ ∂z L ( z, t ; w ) = 2( f Re − z ) − f Im − z ) + 2 t ( | f ( w ) − z | + t ) . In particular, ∆ H K h L ( z, t ; w ) = [ K : R ]( d + 1) . Proof.
Write | f ( w ) − z | = ( f Re − z ) + ( f Im − z ) , where we note that there is no f Im − z term inthe case K = R . The formulas for the derivatives follow directly from basic calculus differentiationrules. We omit the first derivative calculations, and compute ∂ ∂t L ( z, t ; w ) and ∂ ∂z L ( z, t ; w ): ∂ ∂t L ( z, t ; w ) = ∂∂t ∂∂t L ( z, t ) = ∂∂t (cid:18) t | f ( w ) − z | + t − d + 1 t (cid:19) = 2( | f ( w ) − z | + t ) − (2 t ) ( | f ( w ) − z | + t ) + d + 1 t = 2 | f ( w ) − z | − t ( | f ( w ) − z | + t ) + d + 1 t . ∂ ∂z L ( z, t ; w ): ∂ ∂z L ( z, t ; w ) = ∂∂z (cid:18) − f Re − z ) | f ( w ) − z | + t (cid:19) = 2( | f ( w ) − z | + t ) + 2( f Re − z )( − f Re − z ))( | f ( w ) − z | + t ) = 2 t + 2( f Im − z ) − f Re − z ) ( | f ( w ) − z | + t ) . The case of ∂ ∂z L ( z, t ; w ) is similar.We now apply the definition of the hyperbolic Laplacian. If K = R :∆ H K h L ( z, t ; w ) = t · (cid:18) ∂ ∂z + ∂ ∂t (cid:19) L ( z, t )= t (cid:18) − f Re − z ) + 2 t ( | f ( w ) − z | + t ) + 2 | f ( w ) − z | − t ( | f ( w ) − z | + t ) + d + 1 t (cid:19) . Note that in this case, ( f Re − z ) = | f ( w ) − z | , so that the above formula reduces to ∆ H K h L ( z, t ; w ) = d + 1 = [ K : R ]( d + 1) as claimed. In the case K = C , the expression for the Laplacian is moreinvolved:∆ H K h L ( z, t ; w ) = t (cid:18) ∂ ∂z + ∂ ∂z + ∂ ∂t (cid:19) L ( z, t ; w ) − t ∂∂t L ( z, t ; w )= t (cid:18) f Im − z ) − f Re − z ) + 2 t ( | f ( w ) − z | + t ) + 2( f Re − z ) − f Im − z ) + 2 t ( | f ( w ) − z | + t ) + 2 | f ( w ) − z | − t ( | f ( w ) − z | + t ) + d + 1 t (cid:19) − t · (cid:18) t | f ( w ) − z | + t − d + 1 t (cid:19) = t · | f ( w ) − z | + 2 t ( | f ( w ) − z | + t ) + ( d + 1) − t ( | f ( w ) − z | + t ) + ( d + 1)= 2( d + 1)= [ K : R ]( d + 1) . We next compute the derivatives and hyperbolic Laplacian of P K ( w − z, t ): Lemma 3.2.2.
Write w = w + iw , z = z + iz , where we understand w = z = 0 when K = R . hen ∂∂t P K ( w − z, t ) = [ K : R ] · t [ K : R ] − ( | w − z | − t )( | w − z | + t ) K : R ] ∂∂z P K ( w − z, t ) = [ K : R ] · t [ K : R ] ( w − z )( t + | w − z | ) K : R ] ∂∂z P K ( w − z, t ) = [ K : R ] · t [ K : R ] ( w − z )( t + | w − z | ) K : R ] ∂ ∂t P K ( w − z, t ) = [ K : R ]([ K : R ] + 1) · t [ K : R ]+2 − t [ K : R ] | w − z | ( t + | w − z | ) [ K : R ]+2 ∂ ∂z P K ( w − z, t ) = [ K : R ] (4[ K : R ] + 2) t [ K : R ] ( w − z ) − t [ K : R ] ( w − z ) − t [ K : R ]+2 ( t + | w − z | ) K : R ] ∂ ∂z P K ( w − z, t ) = [ K : R ] (4[ K : R ] + 2) t [ K : R ] ( w − z ) − t [ K : R ] ( w − z ) − t [ K : R ]+2 ( t + | w − z | ) K : R ] In particular, ∆ H K h P K ( w − z, t ) = 0 . Proof.
Here again, the result is straightforward calculus. For t , the first derivative is ∂∂t P K ( w − z, t ) = ∂∂t (cid:18) tt + | w − z | (cid:19) [ K : R ] = [ K : R ] · (cid:18) tt + | w − z | (cid:19) [ K : R ] − · t + | w − z | − t (2 t )( t + | w − z | ) = [ K : R ] · t [ K : R ] − ( | w − z | − t )( t + | w − z | ) K : R ] . The first derivative for z is ∂∂z P K ( w − z, t ) = ∂∂z (cid:18) tt + | w − z | (cid:19) [ K : R ] = [ K : R ] · (cid:18) tt + | w − z | (cid:19) [ K : R ] − · t ( w − z )( t + | w − z | ) = [ K : R ] · t [ K : R ] ( w − z )( t + | w − z | ) K : R ] ;the calculation of the first derivative for z is symmetric. The calculations for the second deriva-tives are tedious but straightforward. The fact that ∆ H K h P K ( w − z, t ) = 0 now follows from thecalculations of the derivatives.In order to compute the hyperbolic Laplacian of R F , we pass the operator ∆ H K h into the integraldefining R F and compute the Laplacian of the integrand, which is a product L ( z, t ; w ) · P K ( w − z, t ).Thus we will require a product formula for the Laplacian (see, e.g. **):∆ H K h ( f g ) = g ∆ H K h f + 2 t ∇ std f · ∇ std g + f ∆ H K h g . (20)We apply this to L ( z, t ; w ) P K ( w − z, t ): 19 roposition 3.2.1. Let z ∈ K and t > , and let γ ( w ) = γ z,t ( w ) = tw + z . The hyperbolicLaplacian of L ( z, t ; w ) P K ( w − z, t ) is given ∆ H K h ( L ( z, t ; w ) P K ( w − z, t )) = [ K : R ] P K ( w − z, t ) (cid:18) ( d −
1) + 4 dt | w − z | + t + 4 || γ − ( f ( w )) , γ − ( w ) || (cid:19) , (21) where we recall that ||· , ·|| is the chordal distance on P ( K ) .Proof. First, recall that ∆ H K h L ( z, t ; w ) = [ K : R ]( d + 1) (Lemma 3.2.1) and ∆ H K h P K ( w − z, t ) = 0(Lemma 3.2.2). Inserting this into (20) yields∆ H K h ( L ( z, t ; w ) P K ( w − z, t )) = [ K : R ]( d +1) P K ( w − z, t )+2 t ∇ std L ( z, t ; w ) ·∇ std P K ( w − z, t ) . (22)We next compute the product of the gradients in the above expression. From Lemma 3.2.1 wefind ∇ std L ( z, t ; w ) = 1 | f ( w ) − z | + t · h− f Re − z ) , − f Im − z ) , − ( d − t − t − ( d + 1) | f ( w ) − z | i , where there is no z component when K = R . Similarly, from Lemma 3.2.2 we have ∇ std P K ( w − z, t ) = [ K : R ] · t [ K : R ] ( | w − z | + t ) K : R ] h w − z ) , w − z ) , t − · ( | w − z | − t ) i . The dot product we are interested in can then be rewritten as ∇ std L ( z, t ; w ) · ∇ std P K ( w − z, t ) = [ K : R ] · P K ( w − z, t ) · | f ( w ) − z | + t ) · ( | w − z | + t ) ~v L · ~v P , (23)where ~v L , ~v P are the vector parts of the respective gradients. Explicitly ~v L · ~v P = − f Re − z )( w − z ) − f Im − z )( w − z ) + t − ( − ( d − t − t − ( d + 1) | f ( w ) − z | )( | w − z | − t ) . We note that2( f Re − z )( w − z ) + 2( f Im − z )( w − z ) = 2Re (( f ( w ) − z ) · ( w − z ))= | f ( w ) − z | + | w − z | − | ( f ( w ) − z ) − ( w − z ) | = | f ( w ) − z | + | w − z | − | f ( w ) − w | . Inserting this into the expression for ~v L · ~v P and simplifying gives ~v L · ~v P = − | f ( w ) − z | − | w − z | + 2 | f ( w ) − w | + t − (cid:0) − ( d − t − t − ( d + 1) | f ( w ) − z | (cid:1) (cid:0) | w − z | − t (cid:1) = − ( d + 1) t − (cid:0) | f ( w ) − z | + t (cid:1) (cid:0) | w − z | + t (cid:1) + 2 | f ( w ) − w | + 2 d ( t + | f ( w ) − z | ) . Inserting this into the expression for the product of the gradients given in (23) and simplifying, wefind ∇ std L ( z, t ; w ) · ∇ std P K ( w − z, t )= [ K : R ] · P K ( w − z, t ) (cid:18) − ( d + 1) t − + 2 d | w − z | + t + 2 | f ( w ) − w | ( | f ( w ) − z | + t )( | w − z | + t ) (cid:19) . H K h ( L ( z, t ; w ) P K ( w − z, t )) =[ K : R ]( d + 1) P K ( w − z, t ) + 2 t ∇ std L ( z, t ; w ) · ∇ std P K ( w − z, t )=[ K : R ] P K ( w − z, t ) (cid:18) − ( d + 1) + 4 dt | w − z | + t + 4 t | f ( w ) − w | ( | f ( w ) − z | + t )( | w − z | + t ) (cid:19) . We are done, noticing that the last term can be manipulated as follows t | f ( w ) − w | ( | f ( w ) − z | + t )( | w − z | + t ) = (cid:12)(cid:12)(cid:12) f ( w ) − zt − w − zt (cid:12)(cid:12)(cid:12) (cid:18)(cid:12)(cid:12)(cid:12) f ( w ) − zt (cid:12)(cid:12)(cid:12) + 1 (cid:19) (cid:16)(cid:12)(cid:12) w − zt (cid:12)(cid:12) + 1 (cid:17) = || γ − ( f ( w )) , γ − ( w ) || . We are now ready to prove Theorem 3.2.1:
Proof of Theorem 3.2.1.
We begin by noting that, in computing ∆ H K h R F , we can pass the Laplacianunder the integral sign appearing in (19) since all of the derivatives of L, P are smooth functionsof ( z, t ) ∈ H K . Thus, we will compute∆ H K h R F ([ γ ]) = Z P ( K ) ∆ H K h L ( z, t ; w ) P K ( w − z, t ) dℓ K π . Here we will use the formula derived in Proposition 3.2.1. Note that R P ( K ) P K ( w − z, t ) dℓ K π = R P ( K ) ( γ − z,t ) ∗ ω K = 1, and Z P ( K ) t | w − z | + t P K ( w − z, t ) dℓ K π = Z P ( K ) | γ − z,t ( w ) | + 1 ( γ − z,t ) ∗ ω K = Z P ( K ) | w | + 1 ω K = 1 . Thus, integrating the expression in (21) against dℓ K π and simplifying gives Z P ( K ) ∆ H K h L ( z, t ; w ) P K ( w − z, t ) ω K ( w ) = [ K : R ] ( d −
1) + 4 Z P ( K ) || f γ ( w ) , w || ω K ( w ) ! , which finishes the proof.The calculations of the preceeding sections are enough to prove Theorem 1.0.1: Proof of Theorem 1.0.1:
The fact that R F is smooth on H K follows from the expression for R F givenin Proposition 2.4.1. Properness follows from Theorem 3.1.1 since d ≥
2; likewise, subharmonicityfollows from Theorem 3.2.1.Note that R F is subharmonic even when d = 1, provided f is not the identity map. However,it need not be proper when d = 1; see Section 5.21 .3 Geometry and Minimizers In this section we give a geometric interpretation to the conjugates f γ for which R F ([ γ ]) is mini-mized, in the case that K = C . Note that, viewing ω C as a probability measure on P ( C ), we candefine both f ∗ ω C and f ∗ ω C . We let ω f := f ∗ ω C + f ∗ ω C . It is a positive measure of total mass d + 1 on P ( C ). Our main result in this section is the following: Theorem 3.3.1.
Let f ∈ C ( z ) , and let γ ∈ SL ( C ) / SU ( C ) be a conjugate for which R F ( · ) isminimized. Then the barycenter of the measure d ω f γ on S is ∈ B C , where d ω f γ is the pullback of ω f γ from P ( C ) to S via stereographic projection. This theorem will be an immediate consequence of the following theorem:
Theorem 3.3.2.
Let f ∈ C ( z ) , and view R F as a function on B C . Then ∇ h R F ( ξ ) = − Z S ζ d ω f γ ( ζ ) , where ξ ∈ B C corresponds to the class [ γ ] ∈ SL ( C ) / SU ( C ) .Remark: While the results above are stated for R F as a function on B C , we will heavily rely onexpressions for R F on H C and SL ( C ) / SU ( C ) in proving this theorem.To begin, we recall the expression for R F derived in Section 2.4 (see (7); the precise expressionsfor ψ, ω z,t and c ( z, t ; f ) appear in Section 2.4): R F ([ γ ]) = d −
12 log t + Z P ( C ) ψ ( α ; z, t ) f ∗ ω z,t − d c ( z, t ; f ) . With this expression, we can explicitly determine a directional derivative of R F : Proposition 3.3.1.
Let ~v ∞ ∈ T B C be the direction towards N = (0 , , ∈ S . Then ∂ ~v ∞ R F (0) = 14 Z S | ζ − N | − | ζ + N | c ω f ( ζ ) . Proof.
The direction ~v ∞ ∈ T can be represented by the path { (0 , , t ) : t > } , where t is viewedas a parameter in the hyperbolic metric on B C . This path, in turn, corresponds to the path ofmatrices { [ η A ] : A > } ∈ SL ( C ) / SU ( C ), where η A = (cid:18) e A/ e − A/ (cid:19) . In particular,lim t → R F ((0 , , t )) − R F ((0 , , t = lim A → R F ([ η A ]) − R F ([id]) A = ∂ A ( R F ([ η A ])) | A =0 . We can compute the latter derivative with the help of (7). Note that R F ([ η A ]) = d − A + Z P ( C ) ψ ( α ; 0 , e A ) f ∗ ω ,e A − d + 12 log c (0 , e A ; f ) . (24)22e easily see that ∂ A (cid:0) d − A (cid:1) = d − , and a straightforward calculation shows that ψ A ( α ; 0 , e A ) := ∂ A ψ ( α ; 0 , e A ) = −| α | e A + | α | . and that the Laplacian of ψ A in the α coordinate is ω A := i π ∂ α ∂ α (cid:18) −| α | e A + | α | (cid:19) dαdα = i π | η − A ( α ) | ) · | η − A ( α ) | − | η − A ( α ) | + 1 dαdα = | η − A ( α ) | − | η − A ( α ) | + 1 ω ,e A . Integrating ω A over P ( C ) gives 0, and we conclude dd c ψ A ( · ; 0 , e A ) = ω A . Lastly, note that ∂ A
12 log c (0 , e A ; f ) = e A | b d | | a d | + e A | b d | = e A | f ( ∞ ) | + e A , which is 0 if f ( ∞ ) = ∞ .We are now ready to compute ∂ A R F . Differentiating (24) in A gives ∂ A R F = d −
12 + Z P ( C ) ψ A ( α ; 0 , e A ) f ∗ ω ,e A + ψ ( α ; 0 , e A ) f ∗ ω A + e A | f ( ∞ ) | + e A = d −
12 + Z P ( C ) ψ ( α ; 0 , e A ) f ∗ ω ,e A + ψ ( α ; 0 , e A ) f ∗ dd c ( ψ A ) + e A | f ( ∞ ) | + e A = d −
12 + Z P ( C ) ψ A ( α ; 0 , e A ) f ∗ ω ,e A + Z P ( C ) f ∗ ψ A ( α ; 0 , e A ) dd c ( ψ ) + e A | f ( ∞ ) | + e A , (25)where in the last step we have used the pullback formula for dd c and then applied integration byparts to move the dd c onto the integrand. Recall that dd c ψ = ω ,e A − δ ∞ , so that Z P ( C ) f ∗ ψ A ( α ; 0 , e A ) dd c ψ = Z P ( C ) f ∗ ψ A ( α ; 0 , e A ) ω ,e A − ψ A ( f ( ∞ ); 0 , e A ) . Inserting this into (25) and simplifying yields ∂ A R F = d + 12 + Z P ( C ) ψ A f ∗ ω ,e A + f ∗ ψ A ω ,e A = d + 12 + Z P ( C ) ψ A ( f ∗ ω ,e A + f ∗ ω ,e A ) , (26)where we now interpret f ∗ ω ,e A , f ∗ ω ,e A as the pullback / pushforward in the sense of measures on P ( C ). We define ω ,e A ; f = f ∗ ω ,e A + f ∗ ω z,e A ; it is a measure of total mass d + 1 on P ( C ).Now, recall that ψ A ( α ; 0 , e A ) = −| α | e A + | α | = −| η − A ( α ) | | η − A ( α ) | . Since the measure ω ,e A ; f has total mass23 + 1, the expression in (26) can be written as ∂ A R F = Z P ( C ) − | η − A ( α ) | | η − A ( α ) | ω ,e A ; f = Z P ( C ) − | η − A ( α ) | | η − A ( α ) | ω ,e A ; f = Z P ( C ) || η − A ( α ) , ∞|| − || η − A ( α ) , || ω ,e A ; f = Z P ( C ) || α, ∞|| − || α, || ( η A ) ∗ ω ,e A ; f . (27)We are now essentially done: recall that ω ,e A = ( η − A ) ∗ ω C = ( η A ) ∗ ω C . It is straightforward tocheck then that ( η A ) ∗ ω z,e A ; f = ( f η A ) ∗ ω C + ( f η A ) ∗ ω C = ω f ηA ; setting A = 0 and inserting this into(27) gives ∂ A R F | A =0 = Z P ( C ) || α, ∞|| − || α, || ω f ( α ) . Pulling this back to S under stereographic projection, and noticing that || z, w || = | ˆ z − ˆ w | yields thestatement in the proposition (here, ˆ z, ˆ w are the pullback to S under stereographic projection). Remark:
Using the previous Proposition, one can give a general formula for the directional deriva-tive in any direction at any point ξ ∈ B C ; the formula is not particularly enlightening, and is notneeded in what follows, so we have omitted it. Lemma 3.3.1.
Let µ be a positive measure of finite volume on S . Then for any a ∈ R , we have Z S | ζ − a | − | ζ + a | dµ ( ζ ) = − (cid:18)Z S ζdµ ( ζ ) (cid:19) · a . In particular, the following are equivalent:1. The conformal barycenter of µ is ∈ R .2. For every a ∈ R , R S | ζ − a | − | ζ + a | dµ ( ζ ) = 0
3. There exist three linearly independent vectors a , a , a ∈ R so that Z S | ζ − a i | − | ζ + a i | dµ ( ζ ) = 0 for i = 1 , , .Proof. Let a ∈ R , and write a = ( x, y, z ). Write ζ = ( ζ x , ζ y , ζ z ) ∈ S , so that | ζ − a | − | ζ + a | = − xζ x − yζ y − zζ z . Integrating against µ yields Z S | ζ − a | − | ζ + a | dµ ( ζ ) = − (cid:18)Z S ζ x dµ ( ζ ) (cid:19) x − (cid:18)Z S ζ y dµ ( ζ ) (cid:19) y − (cid:18)Z S ζ z dµ ( ζ ) (cid:19) z = − (cid:18)Z S ζ dµ ( ζ ) (cid:19) · a , · in the last expression is the dot product in R . The measure µ has conformal barycenter0 if and only if R ζdµ ( ζ ) = 0, which happens if and only if (cid:0)R S ζ dµ ( ζ ) (cid:1) · a = 0 for all a ∈ R . Thisestablishes the equivalence of 1 and 2. To see that 3 is equivalent to 2, notice that for fixed b ∈ R ,the expression b · a = 0 vanishes for all a ∈ R if and only if it vanishes on three linearly independentvectors a , a , a ∈ R .We can now prove Theorem 3.3.2 Proof of Theorem 3.3.2.
We first compute the gradient at ξ = 0 ∈ B C . Let ~v ∈ T B C be a unittangent vector. It determines a unique point a ~v ∈ S , and we can find τ ~v ∈ SU ( C ) sending thepoint (0 , , ∈ S to the corresponding a ~v . The path { t~v : t > } in B C corresponds to the path { [ τ ~v · η A ] : A > } in SL ( C ) / SU ( C ). To compute the directional derivative ∂ ~v R F (0) in the ballmodel is the same as computinglim A → R F ([ τ · η A ]) − R F ([id]) A = lim A → R F τ ([ η A ]) − R F τ ([id]) A = 14 Z P ( C ) | ζ − N | − | ζ + N | d ω f τ ( ζ ) , where the last equality is from Proposition 3.3.1. The change of variables by τ gives Z S | ζ − N | − | ζ + N | d ω f τ ( ζ ) = Z S | τ − ( ζ ) − N | − | τ − ( ζ ) + N | c ω f ( ζ )= Z S | ζ − τ ( N ) | − | ζ + τ ( N ) | c ω f ( ζ ) , where in the last step we have used the invariance of the Euclidean distance on R under rotations.Recalling that τ ( N ) = a ~v and applying Lemma 3.3.1, we find that ∂ ~v R F (0) = 14 Z S | ζ − a ~v | − | ζ + a ~v | c ω f ( ζ ) = − (cid:18)Z S ζ c ω f ( ζ ) (cid:19) · a ~v . This is to say that ∇ h R F (0) = − Z S ζ c ω f ( ζ ) . To compute the gradient at an arbitrary point, we use the transformation formula for the R F :since R F ([ γ ]) = R F γ ([id]), it follows from the above calculations that ∇ h R F ( ξ ) = ∇ h R F γ (0) = − R S ζ d ω f γ ( ζ ), where [ γ ] ∈ SL ( C ) / SU ( C ) corresponds to the point ξ ∈ B C .The proof of Theorem 3.3.1 now follows: Proof of Theorem 3.3.1.
By the equivariance of R F , it is enough to assume that γ = . Since γ isa minimizer of R F , the gradient of R F vanishes; by Theorem 3.3.2 this means − R S ζ c ω f = 0, i.e. c ω f is barycentered.We remark that, in the non-Archimedean setting, the analogue of ω f is the measure δ f := f ∗ δ ζ Gauss + f ∗ δ ζ Gauss , where ζ Gauss is the Gauss point in the Berkovich projective line over a com-plete, algebraically closed non-Archimedean field. There is a notion of barycenter in the non-Archimedean context due to Rivera-Letelier, and one can show that Bary( δ f ) is contained in theset MinResLoc( f ). In particular, when MinResLoc( f ) is a single point (this always happens when d is even, see [18] Theorem 1.1), then MinResLoc( f ) = Bary( δ f ). We expect in the complex settingthat Min( f ) is always a single point, and moreover that Min( f ) = Bary( ω f ).25 .4 R F and the Projective Capacity In this section, we give a description of the function R F in terms of Alexandrov’s projective capacity.Let K ⊆ C be compact, and for a function h : K → C let || h || K = sup z ∈ K | h ( z ) | be the sup normon K . Define m k ( K ) = inf (cid:26) || Q || K : Q ∈ C [ X, Y ] homogeneous of degree k, Z S log | Q | dσ = k Z S log | z | dσ (cid:27) . The normalization condition on Q is a multi-dimensional analgue of saying that Q is monic. Anexplicit calculation shows that R S log | z | dσ = . The projective capacity is defined to bepcap( K ) := lim k →∞ m k ( K ) /k ;the fact that this limit always exists is explained in [1]. The following Theorem, whose proof willoccupy the remainder of this section, shows that R ( F ) measures the distortion of pcap( S ) inducedby F : Theorem 3.4.1.
Let f ∈ C ( z ) and let F be any homogeneous lift. Then R ( F ) = d log (cid:18) pcap( S )pcap( F − ( S )) (cid:19) . To prove this theorem, we will utilize several extremal functions; the first two are due to Siciak[20], and the third is the usual pluricomplex Green’s function:Ψ K ( z ) := sup n | Q ( z ) | Q : Q ∈ C [ X, Y ] homogeneous , || Q || K ≤ o Φ K ( z ) := sup n | Q ( z ) | Q : Q ∈ C [ X, Y ] , || Q || K ≤ o V K ( z ) := sup { u ( z ) : u pluri-subharmonic , u − log || z || = O (1) as || z || → ∞ , and u | K ≤ } . Note that Ψ K ( λz ) = | λ | Ψ K ( z ) for all λ ∈ C . As an example, when K = S , we find Ψ S ( z ) = || z || .It is not surprising that there are a number of relationships between these functions; the followingare well-known: Proposition 3.4.1.
Let K ⊆ C be compact, and let Ψ K , Φ K , V K be as above. Then1. V K ( z ) = log Φ K ( z )
2. If K is circled – i.e. w ∈ K ⇐⇒ e iθ w ∈ K ∀ θ ∈ R – then Φ K ( z ) = max { , Ψ K ( z ) } .3. For any homogeneous polynomial F ∈ C [ X, Y ] of degree d , the pluricomplex Green’s functionsatisfies V K ( F ( z )) = dV F − ( K ) ( z ) .Proof.
1. See Klimek’s book [16], Theorem 5.1.7.2. See [20] Section 9, Theorem 3.3. See Klimek’s book [16], Theorem 5.3.1.Combining these yields the following transformation formula for Ψ K :26 orollary 3.4.1. If K ⊆ C is compact and circled, and if F ∈ C [ X, Y ] , then Ψ K ( F ( z )) = Ψ F − ( K ) ( z ) d . Proof.
Note that the homgeneity of F implies that K is circled if and only if F − ( K ) is circled.Combining (1) and (2) of Proposition 3.4.1 gives V K ( z ) = log max { , Ψ K ( z ) } ; since V K ( z ) =log || z || + O (1) as || z || → ∞ , we can choose M so that V K ( z ) > || z || ≥ M . Thus, V K ( z ) = log Ψ K ( z ) for all || z || ≥ M .For any z ∈ C , choose λ so that | λ | · || F ( z ) || > M and | λ | /d · || z || > M . Thenlog Ψ K ( F ( z )) = log (cid:18) | λ | Ψ K ( λ · F ( z )) (cid:19) = − log | λ | + V K ( λ · F ( z ))= − log | λ | + V K ( F ( λ /d z )) = ( ∗ ) . Applying the transformation formula for V K given in Proposition 3.4.1 (3), we see( ∗ ) = − log | λ | + dV F − ( K ) ( λ /d z )= − log | λ | + d log Ψ F − ( K ) ( λ /d z )= log Ψ F − ( K ) ( z ) d . Finally, we will use one last theorem relating the projective capacity to the function Ψ K ( z ): Proposition 3.4.2. (See [6]) Let K ⊆ C be compact. Then log pcap( K ) = − − Z S log Ψ K dσ . We are now ready to prove Theorem 3.4.1
Proof of Theorem 3.4.1.
Observe that R ( F ) = Z S log || F ( z ) || dσ = Z S log Ψ S ( F ( z )) dσ . Applying the transformation formula in Corollary 3.4.1 and the identity in Proposition 3.4.2 yields R ( F ) = d Z S log Ψ F − ( S ) ( z ) dσ = − d − d log pcap( F − ( S )) . (28)But from Proposition 3.4.2 and the fact that Ψ S ( z ) = || z || , we see that log pcap( S ) = − .Plugging this into (28) gives R ( F ) = d log pcap( S ) − d log pcap( F − ( S ))which is the equality asserted in the theorem. 27 Asymptotic behavior d n R F n Our goal in this section is to compute the asymptotic limit of d − n R F ( n ) as a function on hyperbolicspace, and to give a geometric interpretation to the limiting function. We will show that Theorem 4.1.1.
The functions d n R F ( n ) , viewed as functions on the hyperbolic ball B C , convergelocally uniformly to h µ f + C F , where µ f is the measure of maximal entropy for f and h µ is averagedBusemann function introduced in Section 2.6. As in previous sections, while this statement is phrased in terms of the ball model B C , we willcarry out different computations in whichever model is particularly convenient, and then transferthese back to the ball model.Recall also that the function h µ f is minimized precisely on the conformal barycenter of µ f .We will show in the following section that the sets Min( f ) converge in the Hausdorff topology toBary( µ f ).Given γ ∈ SL ( C ), write γ = τ · η A · σ for some τ, σ ∈ SU ( K ) and A >
0. By the SU ( C )-invariance of the norm on C , we find that || F γ ( X, Y ) || = || σ − η − A τ − F ( γ · ( X, Y )) || = || η − A · τ − F ( γ · ( X, Y )) || . Applying Lemma 3.1.1 we find that e − A/ || F ( γ · ( X, Y )) || ≤ || F γ ( X, Y ) || ≤ e A/ || F ( γ · ( X, Y )) || . Applying this to the iterates F ( n ) of F yields e − A/ || F ( n ) ( γ · ( X, Y )) || ≤ || ( F ( n ) ) γ ( X, Y ) || ≤ e A/ || F ( n ) ( γ · ( X, Y )) || and so − A d n + d − n Z S log || F ( n ) ( γ · ( X, Y )) || d Vol S ≤ d − n R F n ([ γ ]) ≤ A d n + d − n Z S log || F ( n ) ( γ · ( X, Y )) || d Vol S . (29) Lemma 4.1.1. As n → ∞ , d − n R F ( n ) ([ γ ]) converges locally uniformly on SL ( C ) / SU ( C ) to e H F ([ γ ]) := Z S H F ( γ · ( X, Y )) d Vol S , where H F ( · ) is the homogeneous escape rate function of F .Proof. We first establish the convergence result. This, in essence, follows directly from the definitionof H F : recall that H F ( X, Y ) = lim n →∞ d − n log || F ( n ) ( X, Y ) || , and this convergence is locally uniform on C . The uniformity allows us to pass the limit into theintegrals in (29), and we find that d − n R F ( n ) ([ γ ]) → Z S H F ( γ · ( X, Y )) d Vol S . The fact that this is locally uniform on SL ( C ) / SU ( C ) comes from (29), where we see that theerror term depends only on A = d SL ([ γ ] , [id]) (there is also dependence on F , coming from the limitdefining H F ). 28he escape rate function H F on C descends to give the Green’s function g F ([ x : y ]) := H F ( x, y ) − log || x, y || on P ( C ). The limit function in the preceeding Lemma can be expressedin terms of the hyperbolic harmonic extension H { g F } . Write Z S H F ( γ · ( X, Y )) d Vol S = Z S H F ( γ · ( X, Y )) − log || γ · ( X, Y ) || d Vol S + Z S log || γ · ( X, Y ) || d Vol S . (30)Rewriting the first integral in affine coordinates gives Z S H F ( γ · ( X, Y )) − log || γ · ( X, Y ) || d Vol S = Z P ( C ) g F ( γ ( α )) ω C = H { g F } ( γ ( j )) , where we recall that γ ( j ) ∈ H C is the point corresponding to [ γ ], and in the last equality we haveused the formula for H { g F } given in Proposition 2.5.1. Returning to (30), we now consider thefunction I ([ γ ]) = Z S log || γ · ( X, Y ) || d Vol S . Note that this is a radial function, in that it only depends on d SL ([ γ ] , [id]): writing γ = τ · η A · σ as above, we find that Z S log || γ · ( X, Y ) || d Vol S = Z S log || η A ( X, Y ) || d Vol S . It will be most convenient to treat this as a function on ( B C , d B ). Lemma 4.1.2.
Identifying SL ( C ) / SU ( C ) with B C , so that I : B C → R , we find ∆ B C h I = 4 . Proof.
Write I ([ γ ]) = R S log || η A · ( X, Y ) || d Vol S as was noted above. In spherical coordinates on B C , we recall that the hyperbolic Laplacian takes the form∆ B C h = 1 − r r (cid:0) (1 − r ) N + (1 + r ) N + (1 − r )∆ σ (cid:1) , (31)where N = r∂ r and ∆ σ is the angular part of the Euclidean Laplacian. Since I is radial, ∆ σ I = 0.To compute the derivative in the radial direction, we first evaluate the integral defining I ([ γ ])explicitly: note that I ([ γ ]) = Z S log || η A · ( X, Y ) || d Vol S = 12 Z P ( C ) log e A | α | + 1 | α | + 1 ω C − A
2= 12 Z P ( C ) log (cid:0) e A | α | + 1 (cid:1) ω C − A + 12 . Expressing ω C in polar coordinates yields I ([ γ ]) = 12 Z ∞ r log (cid:0) e A r + 1 (cid:1) r dr − A + 12 . I ([ γ ]) = Ae A e A − − A + 12 = 12 (cid:18) A (cid:18) e A + 1 e A − (cid:19) − (cid:19) . (32)Identifying [ γ ] with its image ξ ∈ B C , we find that A = d SL ([ γ ] , [id]) = d B ( ξ,
0) = log (cid:18) | ξ | − | ξ | (cid:19) . Inserting this into (32) and simplifying gives I ( ξ ) = 12 (cid:18) log (cid:18) | ξ | − | ξ | (cid:19) · (cid:18) | ξ | | ξ | (cid:19) − (cid:19) = 12 (cid:18) log (cid:18) r − r (cid:19) · (cid:18) r r (cid:19) − (cid:19) . (33)Finally, putting this into the expression for the hyperbolic Laplacian ∆ B C h given in (31) and simpli-fying gives ∆ B C h ( I ( · )) = 4 , from which the assertion in the lemma follows.To summarize what has been shown so far, by identifying a class [ γ ] ∈ SL ( C ) / SU ( C ) withthe corresponding point ξ ∈ B C , we see that d − n R F ( n ) converges locally uniformly to a functionΓ F ( ξ ) := H { g F } ( ξ ) + I ( ξ )on B C with ∆ B C h Γ F = 4.In Proposition 2.6.1 we showed that for any probability measure µ on S , the function h µ : B C → R given h µ ( z ) := − R S log (cid:16) −| z | | z − ζ | (cid:17) dµ ( ζ ) satisfies∆ B C h h µ = 4 . It follows that Γ F − h µ is a hyperbolic harmonic function for any probability measure µ on S . We will be particularlyinterested in the case that µ = µ f is the (pullback of the) measure of maximal entropy of f . Proposition 4.1.1.
The function Γ F − h µ f is constant on B C . More precisely, Γ F ( ξ ) = h µ f ( ξ ) + Z P ( C ) g F ω C − . Proof.
To show that this function is constant, we will consider the boundary behavior of Γ F − h µ f .Write Γ F ( ξ ) = H { g F } ( ξ ) + I ( ξ )for ξ ∈ B C , and h µ f ( ξ ) = − Z S log (cid:18) − | ξ | | ζ − ξ | (cid:19) dµ f ( ζ ) . F ( ξ ) − h µ f can be writtenΓ F ( ξ ) − h µ f ( ξ ) = (cid:18) H { g F } ( ξ ) − Z S log | ζ − ξ | dµ f ( ζ ) (cid:19) + (cid:18) I ( ξ ) + 12 log(1 − | ξ | ) (cid:19) . We consider separately the two terms appearing here: • By Proposition 2.5.2, since µ f has continuous potentials (see, e.g. [10] Th´eor`eme 3.7.1) wefind that lim ξ → ξ ∈ S Z S log | ζ − ξ | dµ f ( ζ ) = Z S log | ζ − ξ | dµ f ( ζ )Noticing that | ζ − ξ | = 2 || ζ, ξ || (where we are writing e ζ, e ξ for the points in P ( C ) corre-sponding to ζ, ξ ∈ S under stereographic projection), the limiting function in the aboveexpression can also be given Z S log | ζ − ξ | dµ f ( ζ ) = Z P ( C ) log || e ζ, e ξ || dµ f ( ζ ) + log 2The function g F ([ x : y ]) = H F ( x, y ) − log || x, y || introduced above satisfies dd c g F = µ f − ω C on P ( C ) (see [4] Section 1.3), so that Z S log | ζ − ξ | dµ f = Z P ( C ) log || e ζ, e ξ || dd c g F + Z P ( C ) log || e ζ, e ξ || ω C + log 2= Z P ( C ) g F ( δ e ξ − ω C ) + log 2 = g F ( e ξ ) − Z P ( C ) g F ω C + log 2 , (34)where we are using that R P ( C ) log || a, b || ω C ( a ) = 0 for any fixed b ∈ P ( C ) (this can be checkeddirectly with an explicit calculation in local coordinates). Let κ ( F ) := R P ( C ) g F ω C .Since H { g F } is the hyperbolic harmonic extension of g F , it follows that, as ξ → ξ ∈ S ,H { g F } ( e ξ ) → g F ( e ξ ); combining this with (34) gives thatlim B C ∋ ξ → ξ ∈ S (cid:18) H { g F } ( ξ ) − Z S log | ζ − ξ | dµ f ( ζ ) (cid:19) = κ ( F ) − log 2 . (35) • We next consider the expression I ( ξ ) − log(1 − | ξ | ). Using the formula for I ( ξ ) derived in(33), this quantity is I ( ξ ) −
12 log(1 − | ξ | ) = 12 (cid:18) log (cid:18) r − r (cid:19) · (cid:18) r r (cid:19) − (cid:19) + 12 log(1 − r ) , where r = | ξ | . Simplifying this expression gives I ( ξ ) −
12 log(1 − | ξ | ) = 12 log(1 + r ) (cid:18) (1 + r ) r (cid:19) −
12 log(1 − r ) (cid:18) ( r − r (cid:19) − . The limit as r → B C ∋ ξ → ξ ∈ S (cid:18) I ( ξ ) −
12 log(1 − | ξ | ) (cid:19) = (log 2) − . (36)31ombining (35) and (36) we find thatlim B C ∋ ξ → ξ ∈ S Γ F ( ξ ) − h µ f ( ξ ) = κ ( F ) − Z P ( C ) g F ω C − . Since Γ F − h µ f is a harmonic function that is constant at the boundary of B C , it must be constantthroughout B C , equal to its boundary value. Thus, Γ F − h µ f = R P ( C ) g F ω C − h µ f is minimized on the conformal barycenter of µ f (viewed as a measure on S viapullback under stereographic projection), we find that Corollary 4.1.1.
The limiting function Γ F is minimized on the conformal barycenter of the mea-sure of maximal entropy µ f . In this section, we prove the following theorem concerning the sets Min( f n ): Theorem 4.2.1.
The sets
Min( f n ) converge in the Hausdorff topology to the barycenter Bary( µ f ) of the measure of maximal entropy µ f . This theorem should not be surprising. The functions R F n converge locally uniformly to h µ f + C F for an explicit constant C F (see Theorem 4.1.1), the sets Min( f n ) are the minimizers for R F n ,and Bary( µ f ) is the minimizer of h µ f .However, it’s important to note that uniform convergence is not, in general, enough to ensurethe Hausdorff convergence of minimizing sets. The extra ingredient needed in our context is givenby the following result: Proposition 4.2.1.
There exists
R > so that the sets Min( f n ) ⊆ B R (0) for all n ∈ N .Proof. Transferring Theorem 3.1.1 to the ball model and applying it to the iterate F n yields d n − d B ( ζ,
0) + log C ( F n ) ≤ R F n ( ζ ) ≤ d n + 12 d B ( ζ,
0) + log C ( F n ) (37)Note that C ( F ) || F n − ( X, Y ) || d ≤ || F n ( X, Y ) || = || F ( F n − ( X, Y )) || ≤ C ( F ) || F n − ( X, Y ) || d for any || X, Y || = 1. Applying this inductively, we find C ( F ) dn − d − ≤ || F n ( X, Y ) || ≤ C ( F ) dn − d − for all n and all || X, Y || = 1; in particular, C ( F ) dn − d − ≤ C ( F n ) and C ( F n ) ≤ C ( F ) dn − d − . Inserting this into (37) yields d n − d B ( ζ,
0) + d n − d − C ( F ) ≤ R F n ( ζ ) ≤ d n + 12 d B ( ζ,
0) + d n − d − C ( F ) . (38)32hus, the largest that R F n (0) can be is d n − d − log C ( F ). But note that, if d B ( ζ, > d − C ( F ) C ( F ) := R then (38) implies R F n (0) ≤ d n − d − C ( F ) < d n − d B ( ζ,
0) + d n − d − C ( F ) ≤ R F n ( ζ ) . In particular, R F n ( ζ ) is not a minimizer of R F n . Hence, R F n must be minimized on B R (0), i.eMin( f n ) ⊆ B R (0). Since R is independent of n , we see that Min( f n ) ⊆ B R (0) for all n ∈ N asclaimed.We also make use of the following basic fact from analysis Lemma 4.2.1.
Suppose f n is a sequence of functions on a metric space ( X, d ) that converges locallyuniformly to a continuous function f on X . If x n ∈ X is a sequence of minimizers of f n , i.e. forany y ∈ X we find f n ( y ) ≥ f n ( x n ) , and if x n → ˆ x ∈ X , then ˆ x is a minimizer of f . We are ready to prove Theorem 4.2.1:
Proof of Theorem 4.2.1.
Since Bary( µ f ) = { y } is a single point, it’s enough to show that for every ǫ >
0, can choose N so that Min( f n ) ⊆ B ǫ ( y ) for all n ≥ N . Suppose this is not the case; thenthere exists ǫ > x n k ∈ Min( f n k ) with x n k B ǫ ( y ) for all k ∈ N .By Proposition 4.2.1, the x n k all lie in some fixed B R (0), hence we can extract a convergentsubsequence again written x n k → ˆ x . By Lemma 4.2.1, the limit ˆ x is a minimizer of the limitingfunction Γ F ( · ) = h µ f ( · ) + C F of the functions d n R F n ; in particular, ˆ x ∈ Bary( µ f ) = { y } , i.e. ˆ x = y .But this is a contradiction, since d B ( x n k , y ) > ǫ for all k . d = 1 In this section we explicitly compute the set Min( f ) and the min-invariant for rational maps f ∈ K ( z ) of degree 1, where K can be R or C . Up to conjugacy, any map f ∈ K ( z ) with degree d = 1can be written in one of the following three forms; note that the lifts have been chosen so that F ( X, Y ) = M · ( X, Y ) ⊤ for the appropriate M ∈ SL ( K ):1. f ( w ) = w , with lift F ( X, Y ) = (
X, Y );2. f ( w ) = w + 1 with lift F ( X, Y ) = ( X + Y, Y ), or3. f ( w ) = λw for λ ∈ K \ { , } , with F ( X, Y ) = ( λ / X, λ − / Y ).In each case, F ( X, Y ) = A · ( X, Y ) ⊤ for some matrix in A ∈ SL ( K ); a direct calculation showsthat | Res( F ) | = | det( A ) | = 1. Thus, to compute m K ( f ) it suffices to determine the minimumvalue of R F for the particular lifts given above. Lemma 5.0.1.
Let f ( w ) = aw + b and let γ ( w ) = tw + z . Then cosh( d H ( f γ ( j ) , j )) = 1 + (cid:12)(cid:12)(cid:12) f ( z ) − zt (cid:12)(cid:12)(cid:12) + ( | a | − | a | . roof. This is a straightforward calculation. Let f ( w ) = aw + b , and γ ( w ) = tw + z . Then f γ ( w ) = aw + f ( z ) − zt , from which we find that f γ ( j ) = | a | j + f ( z ) − zt . By (4), we have cosh( d H ( f γ ( j ) , j )) = 1 + (cid:12)(cid:12)(cid:12) f ( z ) − zt (cid:12)(cid:12)(cid:12) + ( | a | − | a | . (39)Let f ( w ) = aw + b , and let F ( X, Y ) = √ a b √ a √ a ! · ( X, Y ) ⊤ be the homogeneous lift normalizedto have determinant 1. The conjugate f γ can be lifted to a map F γ ( X, Y ) = M γ · ( X, Y ) ⊤ ; recall that d H ( f γ ( j ) , j ) computed in Lemma 5.0.1 is precisely the exponent A appearing in the decomposition M γ = τ · (cid:18) e A/ e − A/ (cid:19) · σ where τ, σ ∈ SU ( K ). In Lemma 2.4.1 we saw that R ( M γ ) = Z S log || M γ · ( X, Y ) ⊤ || d Vol S = log(1 + e A ) − A − log 2 , K = R − + A (cid:16) e A +1 e A − (cid:17) , K = C . The following lemma says that R ( M γ ) is increasing as a function of A = d H ( f γ ( j ) , j ): Lemma 5.0.2.
The functions x log(1 + e x ) − x and x x coth( x ) are increasing on (0 , ∞ ) .Proof. For the first function, we compute ∂ x (cid:16) log(1 + e x ) − x (cid:17) = e x − e x + 1 , which is strictly positive for x ∈ (0 , ∞ ). For the second function, note x coth( x ) = x (cid:16) e x +1 e x − (cid:17) ,whose derivative in x is ∂ x ( x coth( x )) = e x − xe x − e x − . The numerator can be expanded as a Taylor series e x − xe x − ∞ X n =1 (cid:18) n n ! − n +1 ( n − (cid:19) x n ;note that the general term in this series can be simplified to n − n n +1 n ! ; for n = 1 , n ≥ x > ∂ x ( x coth( x )) > , ∞ ).We are now ready to compute Min( f ) and the min-invariant for maps of degree d = 1. Wefollow the cases outlined above: 34. If f ( α ) = α , then R F is constant, so that Min( f ) = H K and m K ( f ) = 1, where F ( X, Y ) =(
X, Y ) is the trivial lift of f to SL ( K ).2. If f ( α ) is conjugate to α α + 1, then by Lemma 5.0.1 we find thatcosh( d H ( f γ ( j ) , j )) = 1 + 12 t . Note that this is independent of z . As t varies in (0 , ∞ ), the fact that cosh( x ) is monotonicallyincreasing implies that A ( t ) = d H ( f γ ( j ) , j ) is decreasing; consequently, R F ([ γ z,t ]) is alsodecreasing as t varies in (0 , ∞ ). An explicit calculation shows thatlim t → R F ([ γ z,t ]) = ∞ , andlim t →∞ R F ([ γ z,t ]) = 0 . Thus, along the oriented geodesics [ ∞ , z ] in H K , the function R F increases monotonically from0 to ∞ . We conclude that R F attains its minimum at {∞} ∈ ∂ H K , and that its minimumvalue is 0.3. If f ( α ) is conjugate to α λα for λ ∈ K \ { , } , then by Lemma 5.0.1 we havecosh( d H ( f γ ( j ) , j )) = 1 + ( | λ | − | λ | + | λ − | · | z | | λ | t . Note that, if z = 0, then d H ( f γ ( j ) , j ) is constant; hence, R F is constant along the hyperbolicgeodesic [0 , ∞ ], and we can explicitly compute R F ([ γ ]) = R F ([id]) = log(1 + | λ | ) − log | λ | − log 2 , K = R − + log | λ | · (cid:16) | λ | +1 | λ | − (cid:17) , K = C . Otherwise, we argue as in the previous case, noting that as t varies in (0 , ∞ ) we find that A ( t ) = d H ( f γ ( j ) , j ) is decreasing, so that R F ([ γ z,t ]) is also decreasing as t varies in (0 , ∞ ).Now an explicit calculation shows thatlim t → R F ([ γ z,t ]) = ∞ , andlim t →∞ R F ([ γ z,t ]) = R F ([id]) . Thus, we’ve shown that R F attains its minimum value m K ( f ) along the entire geodesic [0 , ∞ ],and off of this geodesic R F is strictly larger than this value. References [1] H. Alexander.
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