THE GEOMETRY OF SOME NATURAL CONJUGACIES IN Cn DYNAMICS

We show that under some simple conditions a topological conjugacy h between two holomorphic self-maps f1 and f2 of complexn-dimensional projective space Pn lifts canonically to a topological conjugacyH between the two corresponding polynomial self-maps of Cn+1, and this conjugacy relates the two Green functions of f1 and f2. These conjugacies are interesting because their geometry is not inherited entirely from the geometry of the conjugacy on Pn. Part of the geometry of such a conjugacy is given (locally) by a complex-valued function whose absolute value is determined by the Green functions for the two maps, but whose argument seems to appear out of thin air. We work out the local geometry of such conjugacies over the Fatou set and over Fatou varieties of the original map.

1. Main theorem.This paper concerns the dynamics of iterates of self-maps of complex n-dimensional projective space, P n .It has been shown that such a map is always of the form f (x) = (f 0 (x) : ••• : f n (x)), where each f i (x) is a homogeneous polynomial of x = (x 0 : ••• : x n ) and all of the polynomials f i are required to have the same degree d, which is referred to as the degree of the self-map.Such self-maps naturally generalize rational self-maps of the Riemann sphere to higher dimensions and, in fact, the one-dimensional case, P 1 , is just the Riemann sphere.
One constructs the Green function G of such a self-map f by first defining a map F : C n+1 → C n+1 by F(x) = (f 0 (x),...,f n (x)), where x = (x 0 ,...,x n ) so that ρ •F = f •ρ where ρ : C n+1 \{0} → P n is the standard projection map.Then one constructs the Green function G : Naturally constructed conjugacies between different rational maps of the Riemann sphere (i.e., those conjugacies arising from holomorphic motions, see [2]) are automatically quasiconformal.In higher-dimensional complex dynamics, it is unknown whether some class of maps will play a role analogous to that played by quasiconformal maps in one dimension.
We show that a topological conjugacy between two holomorphic self-maps of P n lifts in an essentially canonical way up to C n+1 .This provides examples of conjugacies between self-maps in higher dimension which are somehow "natural" conjugacies.We work out the geometry of the lifted conjugacies over the Fatou set and over Fatou varieties.
Formally our result is that given an appropriate topological conjugacy between holomorphic self-maps f 1 : P n → P n and f 2 : P n → P n , there exists a canonical lift to a topological conjugacy between the corresponding (polynomial) lifts F 1 : C n+1 → C n+1 and F 2 : C n+1 → C n+1 of f 1 and f 2 , respectively.The proof is rather long and will be done in sections.The theorem is as follows.
Theorem 1.1.Let f 1 , f 2 be holomorphic self-maps of P n of degree d > 1 and assume h : P n → P n is a homeomorphism for which the diagram commutes (i.e., h conjugates f 1 and f 2 ) and such that h is homotopic to the identity map of P n .Let G 1 and G 2 be the Green functions for f 1 and f 2 , respectively, and let F 1 : C n+1 → C n+1 and F 2 : C n+1 → C n+1 be holomorphic lifts of f 1 and f 2 , respectively, to homogeneous polynomial maps of degree d.Then there is a lift H : C n+1 → C n+1 of h to a homeomorphism of C n+1 which satisfies H(βz) = βH(z) for all β ∈ C and all z ∈ C n+1 and for which the diagram commutes (i.e., H conjugates F 1 and F 2 ).Moreover H is unique up to multiplication by a (d − 1)th root of unity and If h is not homotopic to the identity, then precisely the same result holds with the modification that H(βz) = βH(z).
Before we prove the theorem we make a note about the hypothesis that h is homotopic to the identity.In the case of P 1 this is equivalent to hypothesizing that h is orientation preserving, but this is not true in P n when n is even.The following result, whose proof was supplied by Igor Kriz, shows that every homeomorphism of P n is homotopic either to the identity or to the map (z 0 : Proposition 1.2.A homeomorphism of P n is homotopic to either the identity map or the map (z 0 : Proof.Since P n is the 2n+1 skeleton of P ∞ , then the inclusion P n → P ∞ is a 2n+1 equivalence.Letting [X, Y ] denote the set of homotopy class of maps from X to Y then by Whitehead's theorem the map [P n , P n ] → [P n , P ∞ ] is a bijection.Now P ∞ is the Eilenberg-MacLane space K(Z, 2) and so [P n , P ∞ ] H 2 (P n , Z) = Z.
The bijection [P n , P ∞ ] H 2 (P n , Z) is given by g g * (α) for g ∈ [P n , P ∞ ] where α is the fundamental class of H 2 (P ∞ , Z).The fundamental class α ∈ H 2 (P ∞ , Z) is given by identifying H 2 (P ∞ , Z) Hom(H 2 (P ∞ , Z), Z) and specifying α as the cellular cochain which sends the homology class of the unique 2 cell in P ∞ to 1 ∈ Z.Thus, our bijection [P n , P n ] → H 2 (P n , Z) = Z is given by mapping h ∈ [P n , P n ] to h * (ι * (α)), where ι : P n → P ∞ is the inclusion map.Given this it is an exercise in the standard lore of P n cohomology to show that the identity map of P n corresponds to 1 ∈ Z, the map (z 0 : and any homeomorphism of P n must correspond to either ±1.
We remark that the map (z 0 : is orientation reversing if and only if n is odd.Thus, there are no orientation reversing homeomorphisms of P n if n is even (as can also be seen from the cohomology ring of P n ).
We remark also that if f 1 and f 2 have real coefficients, then both h and h are conjugacies and one of these is necessarily homotopic to the identity.
Proof of the main theorem.We prove the main theorem assuming that h is homotopic to the identity.The case in which h is not homotopic to the identity will be an easy consequence at the end of the proof.
We note that if any conjugacy H : This is because the level sets G 1 = 0 and G 2 = 0 in C n+1 are precisely the points whose forward orbit remain bounded, but which are not attracted to the origin under F 1 and F 2 , respectively.Therefore G 1 (x) = 0 if and only if G 2 (H(x)) = 0.The fact that G 2 •H = G 1 then follows from the fact that H(βx) = βH(x) since for any where the final equality comes from the fact that e −G 1 (x) x is a point where To prove the existence and uniqueness of H requires more work.To guide the reader we outline what will follow.First we will show that the conjugacy h lifts to an automorphism of the tautological line bundle of P n .From this automorphism we induce a homeomorphism We will need to rescale H 1 to obtain a conjugacy H between F 1 and F 2 .This will be done by solving a functional equation.Uniqueness of H up to multiplication by a (d − 1)th root of unity will follow from our solution of the rescaling problem.

Lift of h to an automorphism of the tautological bundle.
We let γ be the tautological line bundle over P n obtained by letting the fiber over x ∈ P n be the complex line in C n+1 which corresponds to x.We let γ * be γ with its zero section removed.Then C n+1 \{0} and γ * are isomorphic as holomorphic C * bundles over P n .
Proposition 1.3 (Husemoller).Let ζ be a k-dimensional vector bundle over B, and let f : B 1 → B be a (continuous) map.Then f * (ζ) admits the structure of a vector bundle, and Theorem 1.4 (Husemoller).Let f ,g : B → B be two homotopic maps, where B is a paracompact space, and let ζ be a vector bundle over B .Then f * (ζ) and g * (ζ) are B isomorphic.
We apply Proposition 1.3 to the homeomorphism h : P n → P n to get a line bundle map φ : h * (γ) → γ which is h on the base space.We apply Theorem 1.4 to the line bundle γ and to the homeomorphisms h : P n → P n and id P n .This yields an isomorphism ψ : γ → h * (γ), which is the identity on the base space, between id * P n (γ) = γ and h * (γ).Composing ψ : γ → h * (γ) and φ : h * (γ) → γ yields a line bundle self-map φ • ψ of γ which is h on the base space.

The induced self-map of
Clearly H 1 is a proper, and hence also a closed map.Since it is a closed continuous bijection, it is a homeomorphism.Since H 1 induces the map h on the base space, P n , then ρ( . It easily follows that there is a continuous function s : We want to rescale the homeomorphism H 1 so that it gives a conjugacy between F 1 and F 2 .Thus, given any continuous function λ : The complex line in C n+1 corresponding to z ∈ P n is linearly mapped by H λ to the line corresponding to h(z).The solution H we seek maps the circle G 1 = 0 in the line corresponding to z to the circle G 2 = 0 in the line corresponding to h(z).It follows that G 1 , G 2 , and h specify the desired map H directly, up to the choice of phase angle on each line through the origin.We could easily replace H 1 by an H λ which is correct up to phase angle.We will not do this here because the process we will use to find the phase angle of the correct λ takes care of the norm of λ as well.
Remark 1.5.With regard to the geometry of the conjugacy H, it is this unique phase angle whose meaning is unclear.The uniqueness of this phase angle suggests that H is geometrically interesting.We will be able to resolve this above the Fatou set in Theorem 2.1.However, it is not clear what the geometry of H is like above open subsets of P n which are disjoint from the Fatou set.

The rescaling problem. Straightforward computation yields
There corresponds a unique homeomorphism of the form H λ : for all β ∈ C * .We will prefer to rewrite (1.4) by noting that since P n is simply connected, then for any continuous function λ : P n → C * we can define a continuous function log(λ) : P n → C. Applying log to our equation (λ d /λ • f 1 ) • s≡1 (both sides of which are functions from for some fixed integer n.Subtracting log(s) from both sides gives the functional equation we discussed when we outlined the proof.We will solve it and complete the proof next.

Solution to d • g(z)
− g(f 1 (z)) = q(z).Given a function q : P n → C we need to find a function g : P n → C which satisfies d • g(z) − g(f 1 (z)) = q(z).We define two linear operators on continuous functions g from P n to C by These operators appear also in Ueda [5] on C n+1 rather than P n (according to Ueda they have been used by many authors and in at least one other of his own papers).It is not difficult to see that Ψ maps a continuous function g : P n → C to a continuous function Ψ • g.It is also easy to check that these operators are inverse one to another.Given these operators our problem , then λ = e 2πin/(d−1) • e −Ψ (log(s)) gives all possible solutions λ of our original functional equation (λ d /λ•f 1 )•s≡1.Thus, there is a unique solution λ to the functional equation (1.4) up to multiplication by a (d−1)th root of unity.The corresponding maps H λ are those which satisfy the conclusion of the theorem.This completes our proof of Theorem 1.1 under the assumption that h is homotopic to the identity.Now let c : P n → P n denote the map c(z 0 : Then the results of the main theorem for h follow easily by applying the theorem to the map h 1 .This completes the proof of Theorem 1.1.

Hypothesis is not empty.
We note that Theorem 1.1 does not have an empty hypothesis.To see this we recall definitions and a theorem of Sullivan and McMullen.Definition 1.6.Given a complex manifold X a holomorphic family of rational maps f λ : P 1 → P 1 , is a holomorphic map X × P 1 → P 1 .Definition 1.7.Given a holomorphic family X of rational maps f λ (z), X top ⊂ X is defined to be the set of points α ∈ X such that there is an open neighborhood U ⊂ X about α for which the maps f α and f β are topologically conjugate for all β ∈ U.
The following theorem is proven in [3] (from whence also we take the above two definitions).
Theorem 1.8.In any holomorphic family of rational maps X top is open and dense in X.
Thus, there are many distinct globally topologically conjugate holomorphic self-maps of P 1 .

The geometry of the conjugacies.
Given a holomorphic self-map f : P n → P n of degree greater than one, then for an arbitrary complex variety X a Fatou map r : X → P n is defined to be a holomorphic map such that the sequence of maps f •k • r from X to P n is a normal family.Fatou maps were introduced in [6] and independently in [4].
We will say that a subvariety X ⊂ P n is a Fatou variety if the inclusion map i : X → P n is a Fatou map.The variety X is not required to be a closed subvariety of P n .Fatou components and stable manifolds of hyperbolic periodic points are examples of Fatou varieties.The latter example has the property that its image under a purely topological conjugacy remains a holomorphic curve.It is known that a subvariety X is Fatou if and only if the restriction of the Green function G of f to ρ −1 (X) is pluriharmonic.Another equivalent condition is that each point x of X has an open neighborhood U ⊂ X and a holomorphic map It is easy to verify that the map s U in the last condition is unique up to multiplication by a constant.
Note that if X is a Fatou variety, then the Green function can be locally written as Re g for a holomorphic map g.The fibers of such maps g are independent of the choice of g and form a canonical holomorphic foliation of ρ −1 (X)\{0}.We denote the canonical foliation of ρ −1 (X) \ {0} ⊂ C n+1 \ {0} by ᏸ f (X).Given an open subset U of X, then the leaves of the foliation of ρ −1 (U ) are the images of the maps β•s U : U → C n+1 for β ∈ C * .We will now show that above any Fatou variety X ⊂ P n whose image is a holomorphic variety (and consequently a Fatou variety) the conjugacy H takes a specific geometric form.In particular, H must always map the foliation ᏸ f 1 (X) to the foliation ᏸ f 2 (h(X)), which is interesting since these foliations were defined independently of H. Theorem 2.1.Under the hypothesis of Theorem 1.1, if X and h(X) are Fatou varieties for f 1 and f 2 , respectively, it follows that the map H given by Theorem 1.1 maps the leaves of ᏸ f 1 (X) to the leaves of ᏸ f 2 (h(X)).The restriction of H to any leaf can be locally identified with h.
Proof.Assume that the leaves of ᏸ f 1 (X) are not mapped by H to the leaves of ᏸ f 2 (h(X)).Then we can choose an open subset 1 of a leaf of ᏸ f 1 (X) and a point x ∈ 1 such that H( 1 ) does not lie in the leaf of ᏸ f 2 (h(X)) through H(x).Because the foliations ᏸ f 1 (X) and ᏸ f 2 (h(X)) are invariant under rescaling, we can assume without loss of generality that 1 lies in the zero set of G 1 , and hence H( 1 ) lies in the zero set of G 2 .By shrinking 1 if necessary we can assume that ρ maps 1 biholomorphically onto its image and that 2 is a local plaque of the leaf of Since both 2 and H( 1 ) lie in a level set of G 2 , then ω(z) lies in the unit circle for every z ∈ 1 .Now H(x) lies on 2 so ω(x) = 1.The iterates of F 1 and F 2 are equicontinuous on 1 and 2 , respectively, because the consecutive images of these leaves remain inside the zero sets of G 1 and G 2 and thus remain inside a bounded subset of C n+1 .Choose m > 0 smaller than the distance between the origin and the zero set of G 2 .If ω ≡ 1 on 1 , then given > 0 there exist z 1 ,z 2 ∈ 1 such that ω(z 1 ) = 1, ω(z 2 ) ≠ 1, and such we see that for any such z 1 and z 2 there is some k such that )) > m − .Thus the iterates of F 2 are not equicontinuous on 2 , which is a contradiction.
The final claim in the theorem is immediate from the fact that ρ is a local biholomorphism from the leaves to P n combined with the relationship ρ • H = h • ρ.
To make the geometric meaning of this clear note that above X and h(X) the functions G 1 and G 2 are smooth.Assume that x ∈ ρ −1 (X).Let L 1 be the level set of G 1 through x and let L 2 be the level set of G 2 through H(x).Then L 1 and L 2 are smooth manifolds of real dimension 2n + 1.Let 1 be the leaf of ᏸ f 1 (X) through x and let 2 be the leaf of ᏸ f 2 (H(X)) through H(x).Then the tangent space T x 1 is the unique complex n space lying in T x L 1 and the tangent space T H(x) 2 is the unique complex n space lying in T h(x) L 2 .If H is differentiable at x, then D x H is an R linear map, not a C linear map.Thus, there is no a priori reason to assume that D x H maps T x 1 to T h(x) 2 .D x H must map the tangent space of the radial complex line through x to the tangent space of the radial complex line through H(x).For a generic R linear map of C n+1 that maps a complex line to a complex line (as D x H must do to the tangent space of the radial complex line through x in C n+1 ) there is at most one complex hyperplane in C n+1 not containing which is mapped to a complex hyperplane.Theorem 2.1 affirms that this complex hyperplane is T x 1 which maps to T x 2 .If h is holomorphic, then σ is holomorphic and σ has a removable singularity at ∞.It is easy to show that in this case σ ≡ ω for ω some (d−1)th root of unity.If K(p 1 ) is connected, then the conjugacy h can always be altered to be holomorphic on P 1 \K(p 1 ) (thanks go to the referee and to Mikhail Lyubich for pointing this out).If K(p 1 ) is not connected, then typically h cannot be made holomorphic on P 1 \K(p 1 ).To see this note that if c 1 is a critical point of p 1 which is attracted to ∞, then h(c 1 ) must be a critical point of p 2 which is attracted to infinity.But since σ ≡ ω, then a necessary condition for a holomorphic h to exist is that there is a critical point c

Corollary 2 . 2 .
Under the hypothesis of Theorem 1.1, given any Fatou component U of f 1 and any point x in ρ −1 (U ) \ {0}, there are local holomorphic coordinates about x and H(x) for which H takes the form (u, v) (h(u), v) for u in an open subset of U and v in an open subset of C. 3. Example.Let D denote the unit disk in C and let D denote the disk of radius in C. For any polynomial map p of C there is a Böttcher coordinate b p : U → D for some neighborhood U ⊂ P 1 of ∞ and some > 0. If p has connected Julia set, then b p extends to a biholomorphism from P 1 \ K(p) to D. If p has disconnected Julia set, then b p can be locally defined about any point z of P 1 \ K(p) using a branch of (b p • p •k ) 1/d k .Thus b p is locally well defined up to multiplication by a root of unity.We will abuse notation and refer to such an extension as a local branch of b p .The critical points of b p are z

. 1 )
Independent of whether the Julia set is connected, the function |b p | has a well defined extension |b p | : P 1 \ K(p) → [0, 1) satisfying |b p | • p = |b p | d and |b p (z)| → 1 as z → K(p).The Green function G for p is known to be G(x, y) =      log |y|, x y ∈ K(p), log y b p (x/y) , x y ∈ P 1 \ K(p).

(3. 2 )
Given this description of the Green function, then the leaves of ᏸ p (P 1 \K(p)) are the fibers of the holomorphic function y/b p (x/y) for any local branch of b p .If K(p) has nonempty interior, then the leaves of ᏸ p (K(p)) are horizontal complex lines in C 2 .Given two polynomial maps p 1 and p 2 , a conjugacy h between them, and Böttcher coordinates b 1 and b 2 , respectively, for p 1 and p 2 , then there is a neighborhood V of ∞ for which the function σ(z) ≡ b 2 (h(z))/b 1 (z) is well defined for z ∈ V \ {∞} and satisfies σ • p 1 = σ d .It is straightforward to show that σ extends to a continuous function σ : C \ K(p) → C * satisfying σ • p 1 = σ d .We lift the maps p 1 and p 2 to C 2 , respectively, as P 1 (x, y) = (p 1 (x/y) • y d ,y d ) and P 2 (x, y) = (p 2 (x/y) • y d ,y d ).By the dependence of σ on our choice of the Böttcher coordinate b 2 it follows that making a change of Böttcher coordinate, replacing b 2 with ω • b 2 where ω d−1 = 1, results in replacing σ with ωσ .If K(p) has nonempty interior, there is a choice of Böttcher coordinate b 2 (and corresponding σ ) such that the conjugacy given by Theorem 1.1 takes the form H(x, y) r = lim z→∞ (b 2 (z)/b 1 (z)).If K(p) has empty interior, then σ has a continuous extension to all of C and H(x, y) 2 ∈ C \ K(p 2 ) of p 2 which satisfies b 1 (c 1 ) = ωb 2 (c 2 ) for some choice of local branches of b 1 and b 2 about c 1 and c 2 and some ω a (d − 1)th root of unity.In particular, a holomorphic conjugacy h can not exist unless |b 1 (c 1 )| = |b 2 (c 2 )|.