On parabolic external maps

We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.


Introduction
In this paper we provide a connection between the worlds of real and complex dynamics by proving theorems on degree d ≥ 2 circle coverings which are interesting in the world of real dynamics per se and interesting in the world of complex dynamics through quasi-conformal surgery. The main theorem states that any C 1+BV degree d ≥ 2 circle covering h (where h ∈ C 1+BV means Dh is continuous and of bounded variation), all of whose periodic orbits are weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. Here weakly expanding means that for any periodic point p of period s there exists a punctured neighborhood of p on which Dh s (x) > 1. And metrically expanding means Dh(x) > 1 holds everywhere, except at parabolic points. This theorem strengthens a theorem by Mañé [M] who proved the same conclusion holds under the stronger assumption that h is C 2 with all periodic points hyperbolic repelling.
The real analytic version of the above theorem, which comes for the same price, provides a missing link between the space of parabolic external maps from the theory of parabolic-like maps and metrically expanding circle coverings. For an enlargement on the theory of parabolic-like maps and the role of parabolic external maps in this theory see the introduction to Section 4 and the paper by the first author [L]. Denote by τ the reflection, both in S 1 (τ (z) := 1/z) and in R (τ (z) = z). A parabolic external map is a map h ∈ F d with the following properties: • h : S 1 → S 1 is a degree d ≥ 2 real-analytic covering of the unit circle, with a finite set P ar(h) of parabolic points p of multiplier 1, • the map h extends to a holomorphic covering map h : W ′ → W of degree d, where W ′ , W are reflection symmetric annular neighborhoods of S 1 . We write W + := W \ D, and W ′ + := W ′ \ D, • for each p ∈ P ar(h) there exists a dividing arc γ p satisfying : p ∈ γ p ⊂ W \ D and γ p is smooth except at p, γ p divides W and W ′ into Ω p , ∆ p and Ω ′ p , ∆ ′ p respectively, all connected, and such that h : ∆ ′ p → ∆ h(p) is an isomorphism and D ∪ Ω ′ p ⊂ D ∪ Ω p , -calling Ω = p Ω p and Ω ′ = p Ω ′ p , we have Ω ′ ∪ D ⊂⊂ W ∪ D.
We denote by P d ⊂ F d the set of parabolic external maps, and by P d,1 ⊂ P d the set of h ∈ P d for which P ar(h) is a singleton z 0 . To emphasize the geometric properties of maps h ∈ P d we shall also write (h, W ′ , W, γ) for such maps, where γ = p γ p , though neither the domain, range or dividing arcs are unique or in any way canonical. An external map for any parabolic-like map belongs to P d (see Section 4.2). Note that the set P d is invariant under conjugacy by a real analytic diffeomorphism this is, for any h ∈ P d and φ ∈ F 1 : φ • h • φ −1 ∈ P d . It is easy to see that P d ⊂ T d (see Proposition 4.2). In particular P d,1 ⊂ T d,1 and so any two maps h 1 , h 2 ∈ P d,1 are topologically conjugate by a unique orientation preserving homeomorphism sending parabolic point to parabolic point. Set h d (z) = z d +(d−1)/(d+1) (d−1)z d /(d+1)+1 , and define H d,1 = {h ∈ F d | h ∼ qs h d } (where h ∼ qs h d means that h is quasi-symmetrically conjugate to the map h d ). It is rather easy to see that h d ∈ P d,1 , see Lemma 4.2. Moreover clearly h d ∈ T d so that H d,1 ⊆ T d,1 . In Section 4.3 we prove: Proposition 2.3. Suppose h 1 , h 2 ∈ P d are topologically conjugate by an orientation preserving homeomorphism φ, which preserves parabolic points. Then φ is quasi-symmetric. In particular P d,1 ⊆ H d,1 ⊆ T d,1 .
Let F d := F d /F 1 denote the set of conjugacy classes of maps in F d under real analytic diffeomorphisms, and call π d : F d → F d the natural projection. As a consequence of the above we have (see also page 18): Aknowledgment. The first author would like to thank the Fapesp for support via the process 2013/20480-7.

Topologically expanding self coverings of the unit circle
Recall that for each integer d ≥ 2, the set T * d denotes the collection of all orientation preserving covering maps h : S 1 → S 1 with the following properties: 1. h has degree d; 2. h is a C 1 local diffeomorphism and the derivative Dh has bounded variation; 3. If p is a periodic point of h with period s, then there is a neighborhood U(p) of p such that Dh s (x) > 1 holds for all x ∈ U(p) \ {p}.
Theorem 3.1. For each map h ∈ T * d , Par(h) is finite. Moreover, there exists a positive integer N and a real analytic function ρ : In particular, the theorem claims that a map h ∈ T * d without neutral cycles is uniformly expanding on the whole phase space S 1 , a result proved by Mañé [M] under a stronger assumption that h is C 2 . Some partial result on the validity of Mañé's theorem under the C 1+BV condition was obtained in [MJ].
Recall that a map h ∈ T * d is called metrically expanding if Dh(x) > 1 holds for x ∈ S 1 \ Par(h).
Proof. Let ρ and N be given by Theorem 3.1 and set which is a continuous function with bounded variation. Then a computation shows that which is strictly greater than one for x ∈ S 1 \ Par(h). Let us complete the proof.
The rest of the paper is devoted to the proof of Theorem 3.1. The condition that Dh has bounded variation is used to control the distortion. Recall that the distortion of h on an interval J ⊂ S 1 is defined as Lemma 3.1. There exists a C 0 > 0 such that, for any interval J ⊂ S 1 and n ≥ 1, if J, h(J), .., h n−1 (J) are intervals with pairwise disjoint interiors, then Proof. Since h is a C 1 covering and Dh has bounded variation, log Dh also has bounded variation. For each x, y ∈ J, is bounded from above by the total variation of log Dh.

Proof of Theorem 3.1
The main step is to prove that a map h ∈ T * d has the following expanding properties.
Proposition 3.3. For each h ∈ T * d the following properties hold: (a) Par(h) is a finite set.
(b) There exists a constant K 0 > 0 such that Dh k (x) ≥ K 0 holds for each x ∈ S 1 and k ≥ 1.
(c) For each x ∈ ∞ k=0 h −k (Par(h)), Dh n (x) → ∞ as n → ∞. (d) Let p be a fixed point and let δ 0 > 0, K > 0 be constants. Then there (e) For any K > 0, there exists a positive integer n 0 such that for each n > n 0 and Assuming the proposition for the moment, let us complete the proof of Theorem 3.1. Replacing h by an iterate if necessary, we may assume that all points in Par(h) are fixed points (since Par(h) is finite). We say that a function ρ : S 1 → (0, ∞) is admissible if the following properties are satisfied: (A1) there is δ 0 > 0 such that whenever x ∈ B(p, δ 0 ) \ {p} for some p ∈ Par(h), we have ρ(h(x)) > ρ(x); (A2) for any x ∈ S 1 \ Par(h) and s ≥ 1 with h s (x) ∈ Par(h), we have Lemma 3.2. There exists a real analytic admissible function ρ.
so that φ(x j ) = y j for each j. Then for δ = mǫ/n 2 and for each j : |y j −x j | < δ the map φ is the desired diffeomorphism.
Fix an admissible function ρ as above and let Note that |Dh k (x)| ρ ≥ ηDh k (x) holds for any x ∈ S 1 and any k ≥ 1. We say that a set U is eventually expanding if there exists a positive integer k(U) such that whenever k ≥ k(U) and x ∈ U \ Par(h), we have |Dh k (x)| ρ > 1. The assertion of Theorem 3.1 is that S 1 is eventually expanding.
Completion of proof of Theorem 3.1. By compactness, it suffices to show that each x 0 ∈ S 1 has an eventually expanding neighborhood U(x 0 ).

Geometric expanding properties of topological expanding maps
This section is devoted to the proof of Proposition 3.3. Throughout, fix h ∈ T * d . We shall first establish lower bounds on the derivative of first return maps to small nice intervals.

Recall an open interval
For each x ∈ D(A), the first entry time k(x) is the minimal positive integer There is an arbitrarily small nice interval around any point z 0 ∈ S 1 . Indeed, let O be an arbitrary periodic orbit such that h k (z 0 ) ∈ O for all k ≥ 0. Then for any n, any component of By [MMS], h has no wandering interval which implies that h −n (O) is dense in S 1 . The statement follows.
Proof. Let s 0 be the period of p. Let B 0 ∋ p be an arbitrary nice interval such that B ∩ orb(p) = {p}. For each n ≥ 1, define inductively B n to be the component of h −s 0 (B n−1 ) which contains p. Then B n is a nice interval for each n and |B n | → 0 as n → ∞. Let Since h has no wandering intervals, ε n → 0 as n → ∞. Let δ 0 be the minimum of the length of the components of B 0 \B 1 . Choose n large enough such that • ε n ≤ e −2C 0 δ 0 /K; (where C 0 is the total variation of log Dh.) • Dh s 0 > 1 on B n+1 \ {p} (according to the third property defining T * d ) . Let us verify that A := B n satisfies the desired properties. So let x ∈ A \ A ′ = B n \ B n+1 and let k ≥ 1 be the first return time of x into A. We need to prove that Dh k (x) ≥ K.
To this end, let T be the component of B n \ B n+1 which contains x and let J be the component of Lemma 3.5. For any K ≥ 1, there exists s 0 such that if p is a periodic point with period s ≥ s 0 then Dh s (p) ≥ K. In particular, Par(h) is finite.
Proof. Let p 0 be an arbitrary fixed point of h and for each n = 1, 2, . . ., let Then ε n → 0 as n → ∞.
Now assume that orb(p) ∩ A = ∅. Let I be an open interval bounded by p 0 and some point p ′ in orb(p) with the property that I ∩ orb(p) = ∅. Then I is a nice interval and |I| ≥ δ. Let J be a component of h −s (I) which has p ′ as a boundary point. Then h j (J) ∩ I = ∅ for j = 1, 2, . . . , s − 1 and |J| ≤ ε s . By Lemma 3.1, we have This proves the first statement. As fixed points of h n are isolated for each n ≥ 1, it follows that Par(h) is finite.
Lemma 3.6. For each h ∈ T * d , there exists a constant λ 0 > 1 such that for any x ∈ S 1 \ Par(h), if A is a sufficiently small nice interval containing x, then DR A ≥ λ 0 holds on D(A) ∩ A.
Now let x ∈ S 1 \ Par(h) and let A ∋ x be a nice interval such that |A| < δ and A∩Par(h) = ∅. Now consider y ∈ A with k ≥ 1 as the first return time of y to A. Let J be the component of h −k (A) which contains y. Then h k : J → A is a diffeomorphism with distortion bounded by C 0 . Since J ⊂ A, there is a fixed point p of h k in J. Note p ∈ Par(h). Since h j |J is monotone increasing for all 0 ≤ j ≤ k, k is equal to the period of p. If k ≤ n 0 then Df k (p) ≥ λ 1 and since |J| ≤ |A| < δ, we have Df k (y) ≥ Df k (p) − (λ 1 − λ 0 ) ≥ λ 0 . If k > n 0 , then Dh k (p) ≥ 2e C 0 , and hence Dh k (y) ≥ e −C 0 Dh k (p) ≥ 2 > λ 0 .
Proof of Proposition 3.3. (a). This property was proved in Lemmas 3.5.
(b). By Lemmas 3.4 and 3.6, for any y ∈ S 1 there is a nice interval A(y) ∋ y such that the derivative of the first return map is at least 1. By compactness, there exist y 1 , y 2 , . . . , y r ∈ S 1 such that r i=1 A(y i ) = S 1 . Now consider an arbitrary x ∈ S 1 and k ≥ 1. Define a sequence {i n } ⊂ {1, 2, . . . , r} and {k n } as follows. First let k 0 = −1, take i 0 such that x ∈ A(y i 0 ) and let k 1 = max{1 ≤ j ≤ k : h j (x) ∈ A(y i 0 )}. If k 1 = k then we stop. Otherwise, take i 1 ⊂ {1, 2, . . . , r} \ {i 0 } be such that h k 1 +1 (x) ∈ A(y i 1 ) and let k 2 = max{k 1 < j ≤ k : h j (x) ∈ A(y i 1 )}. Repeat the argument until we get k n = k. Then n ≤ r and Dh k j+1 −k j −1 (h k j +1 (x)) ≥ 1. It follows that This proves the property (b). (c). Assuming h k (x) ∈ Par(h) for all k ≥ 0, let us prove that Dh k (x) → ∞ as k → ∞. By (b), it suffices to show that lim sup k→∞ Dh k (x) = ∞. Let y ∈ ω(x) \ Par(h) and consider a small nice interval A containing y for which the conclusion of Lemma 3.6 holds. Since y ∈ ω(x) there exist n 1 < n 2 < · · · such that h n k (x) ∈ A. By Lemma 3.6, Dh n k+1 −n k (h n k (x)) ≥ λ 0 > 1 for all k.
The proof repeats part of the proof of Lemma 3.4. Let B 0 be a nice interval such that B 0 ⊂ B(p, δ 0 ), B 0 ∩ orb(p) = {p}. Define B n to be the component of h −n (B 0 ) which contains p. Let τ > 0 be the minimal length of the components of B 0 \ B 1 . Given K > 0 let n 0 be so large that (e). Without loss of generality, we may assume that all periodic points in Par(h) are fixed points. Let X 0 = Par(h) and for n ≥ 1, let X n = h −n (Par(h)) \ h −n+1 (Par(h)). So for each y ∈ X n , n is the minimal integer such that h n (y) ∈ Par(h).
Let δ 0 > 0 be a small constant such that h| B(p,δ 0 ) is injective and B(p, δ 0 )∩ Par(h) = {p} for each p ∈ Par(h). Note that this choice of δ 0 implies the following: if y ∈ B(p, δ 0 ) ∩ X m for some m ≥ 1, then max m j=1 d(h j (y), p) ≥ δ 0 . Thus by (d), there is a constant δ > 0 with the following property: if y ∈ B(p, δ) ∩ X m for some m ≥ 1, then Dh m (y) ≥ K/K 0 . Now for each p ∈ Par(h), fix a nice interval A p ∋ p such that A p ⊂ B(p, δ). Given x ∈ X n with n ≥ 1, we shall estimate Dh n (x) from below. Let p = f n (x).
Case 2. Assume now that h j (x) ∈ B(p, δ) for all 0 ≤ j < n. Then n is the first entry time of x into A p . Let J be the component of h −n (A p ) which contains x. Then J, h(J), . . . , h n−1 (J) are pairwise disjoint. By Lemma 3.1, Dh n (x) ≥ e −C 0 |A p |/|J|. Provided that n is large enough, |J| is small so that Dh n (x) ≥ K.

Parabolic external maps
In this section we will prove Theorem 2.4, which relates parabolic external maps to topologically expanding maps and to metrically expanding maps, and which completes the theory of parabolic-like maps. We will start by giving an introduction to parabolic-like maps. We will always assume the degree d ≥ 2, if not specified otherwise.

Parabolic-like maps
The notion of parabolic-like maps is modeled on the notion of polynomiallike maps and can be thought of as an extension of the later theory. A polynomial-like map is an object which encodes the dynamics of a polynomial on a neighborhood of its filled Julia set. We recall that the filled Julia set for a polynomial is the complement of the basin of attraction of the superattracting fixed point ∞, and therefore the dynamics of a polynomial is expanding on a neighborhood of its filled Julia set.
A (degree d) polynomial-like mapping is a (degree d) proper holomorphic map f : U ′ → U, where U ′ , U ≈ D and U ′ ⊂ U. The filled Julia set for a polynomial-like map (f, U ′ , U) is the set of points which never leave U ′ under iteration. Any polynomial-like map is associated with an external map, which encodes the dynamics of the polynomial-like map outside of its filled Julia set, so that a polynomial-like map is determined (up to holomorphic conjugacy) by its internal and external classes together with their matching number in Z/(d − 1)Z. By replacing the external map of a degree d polynomial-like map with the map z → z d (which is an external map of a degree d polynomial) via surgery, Douady and Hubbard proved that any degree d polynomial-like map can be straightend (this is, hybrid conjugate) to a degree d polynomial (see [DH]).
On the other hand, in degree 2 a parabolic-like map is an object encoding the dynamics of a member of the family P A (z) = z + 1/z + A ∈ P er 1 (1), where A ∈ C, on a neighborhood of its filled Julia set K A . This family can be characterized as the quadratic rational maps with a parabolic fixed point of multiplier 1 at ∞, and critical points at ±1. The filled Julia set K A of P A is the complement of the parabolic basin of attraction of ∞. So on a neighborhood of the filled Julia set K A of a map P A there exist an attracting and a repelling direction.
The filled Julia set is defined in the parabolic-like case to be the set of points which do not escape Ω ′ ∪ γ under iteration. As for polynomial-like maps, any parabolic-like map is associated with an external map (see [L]), so that a parabolic-like map is determined (up to holomorphic conjugacy) by its internal and external classes. By replacing the external map of a degree 2 parabolic-like map with the map h 2 (z) = z 2 +1/3 z 2 /3+1 , (which is an external map of any member of the family P er 1 (1)(z) = {[P A ]|P A (z) = z + 1/z + A}, as shown in [L]) one can prove that any degree 2 parabolic-like map is hybrid equivalent to a member of the family P er 1 (1) (see [L]).
The notion of parabolic-like map can be generalized to objects with a finite number of parabolic cycles. More precisely, let us call simply paraboliclike maps the objects defined before, which have a unique parabolic fixed point. Then a parabolic-like map is a 4-tuple (f, U ′ , U, γ) where U ′ , U, U ∪ U ′ , ≈ D, U ′ U, f : U ′ → U is a degree d proper holomorphic map with a finite set P ar(f ) of parabolic points p of multiplier 1, such that for all p ∈ P ar(h) there exists a dividing arc γ p ⊂ U , p ∈ γ p , smooth except at p, γ = p γ p , and such that: • it divides U and U ′ in Ω p , ∆ p and Ω ′ p , ∆ ′ p respectively, all connected, and such that f : ∆ ′ p → ∆ f (p) is an isomorphism and Ω ′ p ⊂ Ω p , • calling Ω = p Ω p and Ω ′ = p Ω ′ p , we have Ω ′ ⊂⊂ U.
The filled Julia set for a parabolic-like map (f, U ′ , U, γ) is (again) the set of points that never leave Ω ′ ∪ γ under iteration.

External maps for parabolic-like maps
The construction of an external map for a simply parabolic-like map (f, U ′ , U, γ) with connected filled Julia set K f is relatively easy, and it shows that this map belongs to P d . Indeed consider the Riemann map α : C \ K f → C \ D, normalized by fixing infinity and by setting α(γ(t)) → 1 as t → 0. Setting reflect the sets and the map with respect to the unit circle, and the restriction to the unit circle h : S 1 → S 1 is an external map for f . An external map for a parabolic-like map is defined up to real-analytic diffeomorphism. From the construction it is clear that h ∈ P d,1 . The construction of an external map for a simply parabolic-like map with disconnected filled Julia set is more elaborated (see [L]), and still produces a map in P d,1 . Repeating the costructions handled in [L] for (generalized) parabolic-like maps, one can see that the external map for a degree d parabolic-like map belongs to P d .
On the other hand, it comes from the Straightening Theorem for paraboliclike mappings (see [L]) that a map in P d,1 is the external map for a paraboliclike map (with a unique parabolic fixed point) of same degree (and the proof is analogous in case of several parabolics fixed points and parabolic cycles).
While the space of external classes of polynomial-like mappings is easily characterized as those circle coverings which are q-s.-conjugate to z → z d for some d ≥ 2, this is not the case for parabolic external classes. Theorem 2.4 gives a characterization for these maps.

Proof of Theorem 2.4
The main technical difficulty for proving Theorem 2.4 is to prove the following property for maps in M d : Lemma 4.1. For any h ∈ M d there is a map φ ∈ F 1 such that the mapȟ := φ • h • φ −1 also belongs to M d and in addition for every orbitp 0 ,p 1 , . . .p s = p 0 ∈ Par(Ȟ) say of parabolic multiplicity 2n, the power series developments ofȞ : T → T at the pointsp k , k ∈ Z/sZ, take the form for some fixed polynomialP (i.e.P depends on the cycle, but not on k) with non-zero constant term and degree at most 4n − 1.
We will first prove the Theorem assuming the Lemma. We prove Lemma 4.1 in Section 4.4. The following Proposition proves that M d ⊂ P d .
Proposition 4.1. For every h ∈ M d there exists ǫ 0 > 0 such that for every 0 < ǫ ≤ ǫ 0 the map h has a holomorphic extension (h, W ′ , W, γ) as a parabolic external map with range W ⊆ {z : | log |z|| < ǫ}. In particular M d ⊂ P d and any map which is conjugate to h by φ ∈ F 1 also belongs to P d . orientation preserving homeomorphism φ, which preserves parabolic points. Then φ is quasi-symmetric.
Proof. Let (h i , W ′ i , W i , γ i ), i = 1, 2 be holomorphic extensions with W ′ i and W i bounded by C 1 Jordan curves intersecting γ i transversely. The case h i ∈ P 1 2 is handled in Lomonaco, [L]. The general case is completely analogous, we include the details for completeness. It suffices to construct a quasi-conformal extension, φ : W It is proved in [L] that the arcs γ 1 p amd γ 2 φ(p) are quasi-arcs and that this extension, which is C 1 for z = p, is quasi-symmetric. Next extend φ as a diffeomorphism between the outer boundary of W 1 + and W 2 + respecting the intersections with γ i , i.e. besides being a diffeomorphism it satisfies φ(γ 1 p (±1)) = γ 2 φ(p) (±1). Then φ is defined as a quasi-symmetric homeomorphism from the quasi-circle boundary of ∆ 1 p to the quasi-circle boundary of ∆ 2 φ(p) for each p ∈ Par(h 1 ). We extend φ as a quasi-conformal homeomorphism φ : ∆ 1 p → ∆ 2 φ(p) . Next consider the C 1 lift φ : ∂W ′ 1 → ∂W ′ 2 of φ • h 1 to h 2 respecting the dividing multi arcs. We next extend φ by φ on ∂W ′ 1 ∩ W + 1 . For each i = 1, 2 the connected components of W + i \ W ′ i are quadrilaterals Q i p indexed by the p ∈ Par(h i ) preceding Q i p in the counter-clockwise ordering. Moreover φ thus defined restricts to a piecewise C 1 and hence quasi-symmetric homeomorphism from the boundary of Q 1 p to the boundary of Q 2 φ(p) . Extend this boundary homeomorphism to a quasi-conformal homeomorphism between Q 1 p and Q 2 φ(p) . Call the thus extended map φ 1 and its domain and range U 1 1 and U 1 2 respectively. Define recursively for i = 1, 2 and n ≥ 1: as the quasi-conformal extension of φ n , which on h −1 (i.e. lift of φ n •h 1 to h 2 ). Then φ n ∪φ converges uniformly to a quasi-conformal homeomorphism φ ∞ : W + 1 → W + 2 , which conjugates dynamics except on ∆ ′ 1 . Thus φ is the restriction to S 1 of a quasi-conformal homeomorphism and thus it is a quasi-symmetric map.
Proof. Let h ∈ P d and let (h, W ′ , W, γ) be a degree d holomorphic extension of h as a parabolic external map with dividing multi arc γ and associated sets ∆ ′ and ∆. We shall first redefine Ω and Ω ′ so as to be τ -symmetric : Ω = W \∆ ∪ τ (∆) and Ω ′ = W ′ \∆ ′ ∪ τ (∆ ′ ) then Ω ′′ := h −1 (Ω) ⊂ Ω ′ Ω. It follows that each p ∈ Par(h), say of period n, is admits the circle as repelling directions. Indeed if not then it would have a τ -symmetric attracting petal along S 1 to one or both sides. However since Ω ′′ ⊂ Ω the parabolic basin for h n containing such a petal would be a proper basin and thus would contain a critical point.
To prove that all other periodic orbits are repelling, let ρ denote the hyperbolic metric on Ω. Then each connected component V of Ω ′′ is a subset of U ∩ W ′ for some connected component U of Ω. Thus h is expanding with respect to the conformal metric ρ. Since any non parabolic orbit is contained in Ω ′′ ∩ S 1 it follows that all non parabolic orbits are repelling. This proves the first inclusion. The equality sign is immediate from Corollary 2.2. By Theorem 2.1, any h ∈ T d is real analytically conjugate to a map h ∈ M d , and so by Proposition 4.1 we also have h ∈ P d . So we obtain

Proof of Lemma 4.1
This section is completely devoted to proving Lemma 4.1. Let us start by noticing that it follows from the definition of M d that h only has finitely many parabolic points. The proof of Lemma 4.1 uses the idea of the proof of Corollary 3.2 to recursively construct conjugacies to maps which full-fills the requirements ofȞ to higher and higher orders. It turns out that after two steps of the recursion we arrive at the desired mapȞ and obtain the conjugacy as the composition of the pair of conjugacies from the recursion. The recursion is given by the following procedure: Let h ∈ M d be arbitrary, let N = N h denote the least common multiple of the periods of parabolic orbits for h and let L := (d N − 1)/(d − 1). Define a real analytic diffeomorphism φ : R −→ R and a new real analytic diffeomorphism H (lift of degree d covering h) as follows: with equality if and only if x ∈ Par(H), thus H ∈ M d . For p ∈ Par(H) with period s, set p := φ(p) ∈ Par( H), p k := H k (p), Let 2n > 0 denote the common parabolic degeneracy. A priori the power series developments (Taylor expansions) of H around the points p k could have non-linear terms of order less than 2n + 1. However, since h ∈ M d the leading non-linear term must be of odd order, say 2m + 1 (and have positive coefficient), and Claim 4.1 implies m = n. Write where P k is a polynomial of degree at most (2m − 1), P s+k = P k for k ≥ 0 and where P k (0) > 0 for at least one k, 0 ≤ k < s. Then for each k the Taylor approximation to order 4m of H at p k takes the form is independent of k ≥ 0 and moreover for x close to p k and j ≥ 1: . (6) Let us first see that the Claim implies m = n. Since H and H are analytically conjugate the parabolic degeneracy of H at p is also 2n. However since the coefficient of the leading terms in (4) are non-negative and at least one of them is positive, it follows from (5) that the constant term of P is positive and then from (6) that the degeneracy is 2m. Therefore m = n.
Proof. Towards a proof of the Claim a routine computation and induction shows that for all j ≥ 0 the Taylor series of H j to order 4m at p k is given by: and thus with Q k := (2m + 1) Continuing to compute H ′ (φ(x)) for x near p k starting from the first term of (3) and using From the formula for φ we find the expansion of φ to order 2m at p k : so that the expansion for φ −1 to order 2m at p k is: and thus the expansion for H ′ to order (4m − 1) at p k is: So by integration from p k we find from which the Claim follows, since N is a multiple of s and the terms of the sum are repeated N/s times.
Claim 4.2. Suppose the Taylor expansions of H around the points p k take the form where P and R k are polynomials of degree at most (2n − 1), P with P (0) > 0 is independent of k and R s+k = R k for k ≥ 0. Then for each k the Taylor expansion of H to order 6n at p k takes the form H( x) = p k+1 +( x− p k )(1+( x− p k ) 2n · P ( x− p k )+( x− p k ) 4n · R( x− p k )+O( x− p k ) 6n ), (8) where R and P (x) = L 2n P (Lx) with P (0) > 0 are polynomials of degree at most 2n − 1 and are independent of the point in the orbit of p = φ(p).
Proof. The proof of this Claim is similar to the proof of the first Claim, and we only indicate the differences. For proving a formula for the j-th iterate the following formula is simple and useful P (x(1 + x 2n P (x))) = P (x) + x 2n · x · P ′ (x) · P (x) + O(x) 4n (9) (Note that the term x 2n · x · P ′ (x) · P (x) contains terms of order larger than or equal 4n, but taking them out only complicates the formula.) By induction for each j ≥ 1 and x close to p k we find H j (x) = p j+k + (x − p k )(1 + (x − p k ) 2n · j · P (x − p k ) + (x − p k ) 4n · j(j − 1) 2 ((2n + 1)(P (x − p k )) 2 where F j (x) := x(1 + x 2n · j · P (x) + x 4n · j(j − 1) 2 ((2n + 1)(P (x)) 2 + xP ′ (x)P (x))) is independent of k, i.e. independent of the starting point in the orbit of p. As above define Q by the formula x 2n Q(x) := d dx (x 2n+1 P (x)), and thus Q(x) = (2n+1)P +x·P ′ , and S k by the formula x 4n S k (x) := d dx (x 4n+1 R k (x)), and thus S k (x) = (4n + 1)R k + x · R ′ k = S s+k (x). Then Computing H ′ (φ(x)) from the second formula in (7) we obtain That is the terms of H ′ (φ(x)) depending on k are the terms of order at least 4n. From the definition of φ and (10) we see that φ is independent of k to order 4n and thus the same holds for φ −1 . Combining this with the above shows that H ′ is independent of k to order 6n − 1 and thus H is independent of k up to and including order 6n, as promised by the Claim.