A characterization of Sierpinski carpet Rational maps

In this paper, we prove that a postcritically finite rational map with non-empty Fatou set is Thurstion equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpinski carpet


Theorem 1.1. A postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet.
To verify this theorem, we will recall some basic definitions and results in Section 2, and prove a series of lemmas about homotopy and isotopy in Section 3 and Section 5. The detailed proof of Theorem 1.1 is left in Section 4.
We will end the introduction with two remarks.
1. There are many examples of postcritically finite rational maps with Julia sets homeomorphic to the standard Sierpiński carpet (see e.g. [17,Appendix] and [19]), and these Julia sets has the quasisymmetric rigidity [2]. Conjecturally, the components of these rational maps are relatively compact in the space of rational functions up to Möbius conjugation. [14,Question 5.3] 2. In the proof of the main theorem, we use the following trick: we first construct a homotopy H : S 2 × I → S 2 rel. P between two homeomorphisms, and then modify it to an isotopy relative to P , where P is a finite subset of S 2 . We emphasize that this result is generally false; see Section 3 for a counterexample and detailed discussion.

Notations
The 2-sphere is denoted by S 2 , the Riemann sphere by C and the open unit disk by D. The closure and interior of a subset K ⊂ S 2 is denoted by cl(K) and int(K) respectively. The spherical metric on S 2 is σ = 2|dz| 1+|z| 2 . The set of critical points of a branched covering F is denoted by crit(F ) and the set of postcritical points by post(F ). The Julia set of a rational map f will be denoted by J f ; the Fatou set is F f .

Expanding Thurston maps
Let F : X → Y be a continuous map between two domains X, Y ⊂ S 2 . The map F is called a branched covering if for each point p ∈ X, there exist an integer n ≥ 1, open neighborhoods U of p and V of q := F (p), and orientation-preserving homeomorphisms φ : U → D and ψ : V → D with φ(p) = ψ(q) = 0 such that ψ • F • φ −1 (z) = z n for all z ∈ D.
The integer deg F (p) := n ≥ 1 is called the local degree of F at p. A point c with deg F (c) ≥ 2 is called a critical point of F . A branched covering without critical points is called a covering.
Let F : S 2 → S 2 be a branched covering. The set of critical points of F is denoted by crit(F ), and the postcritical set post(F ) is defined as post(F ) := ∪ n≥1 F n (crit(F )).
The map F is called postcritically finite if #post(F ) < ∞.
Definition 2.1. A Thurston map is an orientation-preserving, postcritically finite, branched covering of S 2 . We fix a base metric ρ on S 2 that induces the standard topology on S 2 . Consider a Jordan curve C ⊃ post(f ). The Thurston map F is called expanding if where mesh F −n (C) denotes the maximal diameter of a component of S 2 \ F −n (C).
It was shown in [1] that the expansionary property is independent of the choice of the Jordan curve C ([1, Lemma 6.1]), and the base metric ρ on S 2 as long as it induces the standard topology on S 2 ([1, Proposition 6.3]).

Partitions of S 2 induced by Thurston maps
Let F : S 2 → S 2 be a Thurston map and fix a Jordan curve C ⊂ S 2 with post(F ) ⊂ C. The closure of one of the two components of S 2 \ C is called a 0-tile (relative to (F, C)). Similarly, we call the closure of one component of S 2 \ F −n (C) an n-tile (for any n ≥ 0). The set of all n-tiles is denoted by X n (C). For any n-tile X, the set F n (X) = X 0 is a 0-tile and F n : X → X 0 is a homeomorphism, (2.1) see [1,Proposition 5.17]. This means in particular that each n-tile is a closed Jordan domain. The definition of "expansion" implies that n-tiles become arbitrarily small. Clearly, for each n ≥ 0, all n-tiles relative to (F, C) form a partition of S 2 .

Postcritically finite Sierpiński carpet rational map
Let f be a postcritically finite rational map. It was known that J f is connected and locally connected ( [15,16] Since the Julia set J f is locally connected, it follows from Carathéodory's theorem that the conformal map η extends to a continuous and surjective map η U : D → U . An internal ray of U is the image η U ([0, 1)θ) for unit number θ ∈ ∂D. Note that internal rays are mapped to internal rays under f .
A set S ⊂ C is called a (Sierpiński) carpet if it is homeomorphic to the standard Sierpiński carpet. By Whyburn's characterization [20], a set S ⊂ C is a carpet if and only if it can be written as S = C \ ∪ n≥1 D n , where all D n are Jordan domains with pairwise disjoint closures, such that the interior of S is empty and the spherical diameters diam σ (D n ) → 0 as n → ∞.
We say a postcritically finite rational map to be a postcritically finite carpet rational map if its Julia set is a carpet. This means that each Fatou component is a Jordan domain and distinct components of the Fatou set have disjoint closures. Furthermore, the boundary of a component of the Fatou set cannot contain postcritical points.
3 Homotopy, isotopy and Thurston equivalent Definition 3.1 (Relative homotopy and isotopy). Let X, Y be topological spaces and A be a subset of X (maybe empty). Let φ, ψ be continuous maps from X to Y . We say that φ and ψ are homotopic rel. A if there exists a continuous map H : If the map H| X×t : X → Y is a homeomorphism for each t ∈ [0, 1], we call H an isotopy rel. A.
Let H : X × [0, 1] → Y be a homotopy. For simplity, we usually denote the map H| X×t :  (2) Let h : ∂D × I → ∂D be the homotopy between id| ∂D and φ| ∂D relative to A with h(·, 0) = id| ∂D , h(·, 1) = φ| ∂D and h(x, t) = x for all x ∈ A, t ∈ I. We obtain the desired homotopy by a small change of the "Alexander trick".
We complementarily define H(z, 0) := z for t = 0 and H(z, 0) := φ(z) for t = 1, then H : D × I → D is a homotopy between id and φ relative to A.
A classical result about the modification of homotopy to isotopy in surfaces is due to D. B. A. Epstein [9]; see also [10,Theorem. 1.12]. One may ask when marking a finite set P in the surface S (defined as in the theorem above), are two orientation preserving homeomorphisms of S that are homotopic rel. ∂S ∪P still isotopic rel. ∂S ∪ P ? The answer is NO in general. Here is a simple counterexample: Choose S = D, the closed unit disk, and let P ⊂ D contain at least two points. Let h be a Dehn twist on D along a Jordan curve surrounding P .
A similar counterexample can be given on S 2 with at least four marked points.
So, in general, two homeomorphisms of an orientable surface homotopic relative to marked points are not necessarily isotopic relative to the marked points. However, if the homotopy is well chosen, the conclusion holds. We leave the proof of the following lemma in the appendix.
Lemma 3.4. Let P be a finite set in an orientable surface S, and H : S × I → S rel. P ∪ ∂S be a homotopy such that H 0 = id S and h := H 1 is an orientation preserving homeomorphism. If each , all but one components of (S \ P ) \ K p are simply-connected, and they are pairwise disjoint, then φ is isotopic to id S rel. P ∪ ∂S.
At the end of this section, we introduce the concept of Thurston equivalent. Definition 3.5 (Thurston equivalent). Two Thurston maps F, G on S 2 are said to be Thurston equivalent if there exist homeomorphisms ψ, φ : S 2 → S 2 that are isotopic rel. post(F ) and satisfy G • ψ = φ • F , that is, the following commutative diagram follows:

A characterization of carpet rational maps
The objective of this section is to prove Theorem 1.1. We will first summarize the idea ( Section 4.1) and then give the detailed proof ( Sections 4.2 and 4.3).

The outline of the proof
Let f be a postcritically finite rational map, and F a Thurston map.
For the necessity, by repeatedly using the isotopy lifting theorem, we obtain a sequence of homeomorphisms {φ n } such that φ n • f = F • φ n+1 and φ n is isotopic to φ n+1 rel. f −n (post(f )) for all n ≥ 0. This sequence of homeomorphisms converges to a semi-conjugacy h from f : C → C to F : S 2 → S 2 by the expansionary property of F . With the properties of this semi-conjugacy h, we can prove that the Julia set of f is a Sierpiński carpet.
The sufficiency proceeds as follows. Suppose that f has Sierpiński carpet Julia set. By collapsing the closure of each Fatou component to a point, we obtain the quotient map π : C → S 2 , by which the rational map f descends to an expanding Thurson map (This assentation is proven in Section 4.2). This yields a semi-conjugacy π from the rational map f to an expanding Thurston map F . We can carefully choose a homeomorphism ψ in the homotopy class of π rel. post(f ) such that ψ has a lift φ along f and F , i.e., F • ψ = φ • f on C, and the homeomorphism φ is homotopic to π rel. post(f ). We then get a homotopy rel. post(f ) between ψ and φ by concatenating the homotopy between ψ, π and that between π, φ. This homotopy turn out to satisfy the properties of Lemma 3.4. It follows that φ and ψ are isotopic rel. post(f ).

The expanding quotient
We will show in this part that any postcritically finite carpet rational map can be semiconjugated to an expanding Thurston map. The base of this fact is Moore's Theorem. (1) the equivalence relation ≡ is closed, (4) there are at least two distinct equivalence classes.
Let f be a postcritically finite rational map with Sierpiński carpet Julia set. We then define an equivalence relation ∼ on C: for any z, w ∈ C, z ∼ w if and only if either z = w or z, w belong to the closure of a common Fatou component.
We claim that the equivalence relation ∼ satisfies the 4 properties of Lemma 4.1. Clearly, the properties (2), (3), (4) holds. To check the property (1), it suffices to show that given two convergent sequences (z n ) n≥1 and (w n ) n≥1 in C with z n ∼ w n for all n ≥ 1 it follows that lim z n ∼ lim w n . This is clear in the case when for sufficiently large n the points z n and w n are contained in some fixed equivalence class, since each equivalence class is compact. Otherwise, we may assume that for distinct n, m ≥ 1 the points z n and z m are contained in distinct equivalence classes. In this case, the diameter of the equivalence class containing z n becomes arbitrarily small as n → ∞ by Lemma 2.2, then we have lim z n = lim w n . Thus ∼ is closed.
Using Lemma 4.1, the quotient space with the quotient topology is homeomorphic to S 2 . We identify C/ ∼ with S 2 so that the quotient map can be written as the continuous map π : C → S 2 . This map is also uniformly continuous with respect to the spherical metric σ.

Since the equivalence relation
for all x ∈ S 2 , that is, the following commutative diagram holds: Furthermore, note that ∼ is also strongly invariant, i.e., the image of any equivalence class is an equivalence class, we then have the following result, see [1, Corollary 13.8] for a proof.
We now remain to show the expansion of F . The following topological result is needed.
Let f be a postcritically finite rational map with Sierpiński carpet Julia set. An arc or a Jordan curve in C is called regulated (with respect to f ) if its intersection with the closure of each Fatou component is either empty or a connected set, i.e., one point or one arc. 2. Let C be a regulated Jordan curve and V 0 , V 1 the two components of C \ C. Then C := π( C) is a Jordan curve, and π(cl( V 0 )), π(cl( V 1 )) are the closures of the two components of S 2 \C.
Proof. 1. Remember that π : C → S 2 is the quotient map defined above. We set We first claim that there exist closed disk neighborhoods D k for each point p k ∈ P such that they are pairwise disjoint and their boundaries avoid E. To see this, notice that S r,k := {x ∈ S 2 | σ(x, p k ) = r} r>0 is a uncountable family of pairwise disjoint sets and E(= π(F f )) is countable. So we may choose sufficiently small r k for each k ∈ {1, . . . , N } such that S r k ,k ∩ E = ∅ and S r i ,i ∩ S r j ,j = ∅ (i = j). The neighborhoods D k are defined as D k := {x ∈ S 2 | σ(x, p k ) ≤ r k }.
We then claim that there are pairwise disjoint open arcs γ 1 , . . . , γ N ⊂ S 2 \ (∪ 1≤k≤N D k ) such that γ k joins D k ,D k+1 with D N +1 := D 1 and γ k ∩ E = ∅ for each k ∈ {1, . . . , N }. Indeed, it is easy to find a sequence of pairwise disjoint open arcs e 1 , . . . , e N ⊂ S 2 \ (∪ 1≤k≤N D k ) such that γ k joins D k and D k+1 . Moreover, for each k we may also choose a small neighborhood A k of e k within S 2 \ (∪ 1≤k≤N D k ) so that A 1 , . . . , A N are still pairwise disjoint. Let h k : A k → C be an injective map with h k (e k ) = (0, 1), the open unit interval. Since h k (A k ) contains an uncountable family of pairwise disjoint horizontal intervals and h k (A k ∩ E) is countable, we may choose a horizontal interval in h k (A k ) disjoint with h k (E) such that its preimage by h k joins D k and D k+1 . We denote this arc by γ k . Then the arcs γ 1 , . . . , γ N satisfy the requirements in the claim. For each k ∈ {1, . . . , N }, set a k and b k the intersection of ∂D k with γ k−1 and γ k respectively.
By the two claims above, the lifts γ k := π −1 (γ k ) (resp. S k := π −1 (∂D k )), k ∈ {1, . . . , N }, are pairwise disjoint open arcs (resp. Jordan curves) in C \ F f . They are therefore regulated. Besides, we also see that the boundary of D k := π −1 (D k ) is exactly S k , and the intersection of ∂ D k and ∪ N k=1 cl( γ k ) are two points a k := π −1 (a k ) and b k := π −1 (b k ). Therefore, to obtain a regulated Jordan curve containing P , it is enough to select a regulated arc α k in each D k that passes through the point p k and joins the points a k , b k ∈ ∂ D k . This can be easily done if one note that each J k := D k ∩ J f is a Sierpiński carpet and it is mapped onto the standard carpet by a self-homeomorphism of S 2 . Finally, the set C := (∪ N k=1 γ k ) ∪ (∪ N k=1 α k ) is a regulated Jordan curve containing the set P .

2.
By the definition of the regulated Jordan curves, the map π collapses one point or one arc on C to a point. It follows from [1, Lemma 13.30] that C is a Jordan curve. Let x = y belong to a component of S 2 \ C. Then π −1 (x) and π −1 (y) are contained in a common component of C \ C. Otherwise, we pick an arc γ in S 2 \ C joining x and y. By [5, Lemma 3.1], the set π −1 (γ) is a continuum containing π −1 (x) and π −1 (y). It hence intersects C. Consequently, we get γ ∩ C = ∅, a contradiction.
By this fact, we can label the two components of S 2 \ C by V 0 and V 1 such that π −1 (V 0 ) ⊂ V 0 and π −1 (V 1 ) ⊂ V 1 . It implies that π(cl( V i )) ⊂ cl(V i ) for i = 0, 1. Since π is surjective, we have π(cl( V 0 )) = cl(V 0 ) and π(cl( V 1 )) = cl(V 1 ) Applying 1 of Lemma 4.3 to the case of P := post(f ), we obtain a regulated Jordan curve passing through post(f ). Fix this curve and denote it by C f . By 2 of Lemma 4.3, the set C F := π(C f ) is a Jordan curve in S 2 containing post(F ). We denote by X n (C f ) and X n (C F ) the set of n-tiles relative to (f, C f ) and (F, C F ) respectively. The following lemma implies a correspondence between them.
Lemma 4.4. For any n ≥ 0, the map Ψ n : X n (C f ) → X n (C F ) defined by Ψ( X n ) → π( X n ) is well-defined and one to one.
Proof. Note that f n and F n are branched covering of degree d n , then the cardinality of X n (C f ) and X n (C F ) are the same, equal to 2d n . For any X n ∈ X n (C f ), by (2.1) the restriction f n : X n → X 0 ∈ X 0 (C f ) is a homeomorphism. It follows that ∂ X n k is a regulated Jordan curve as well. The equation F n • π = π • f n implies that π(f −n (C f )) = F −n (C F ). Then an argument similar to the one used in the proof of 2 of Lemma 4.3 shows that, for each X n k ∈ X n (C F ), the set π −1 (int(X n k )) is contained in a unique n-tile X n τ (k) in X n (C f ) with τ (k) = τ (k ′ ) if k = k ′ , and π( X n τ (k) ) = X n k . This fact gives an one to one correspondence between X n (C f ) and X n (C F ) by X n τ (k) → X n k . The lemma is proved.
Let K 0 be the union of all postcritical Fatou components, i.e., the Fatou components containing the postcritical points of f , and set K n := f −n (K 0 ) for all n ≥ 0.
Lemma 4.5. The maximum of diam σ ( X n \ K n ) with X n ∈ X n (C f ) converges to 0 as n → ∞.
Proof. For any n ≥ 0 and X n k ∈ X n (C f ), we set U n k := int( X n k \ K n ). As ∂ X n k is regulated, it follows that U n k is a Jordan domain. Set X 0 i := f n ( X n k ), then it is an 0-tile and f n : X n k → X 0 i is a conformal homeomorphism. By the definition of K n , we have the conformal homeomorphism The proof goes by contradiction. Without loss of generality, we assume that {U n kn = int( X n kn \ K n )} n≥1 is a sequence of Jordan domains such that diam σ (U n kn ) → C > 0 as n → ∞, (4.2) and f n (U n kn ) = U 0 0 for all n ≥ 1. Consider the sequence of conformal maps {g n := f −n | U 0 0 }. It is a normal family, because K 0 ⊆ K n and hence their images avoid the set K 0 . We then obtain a subsequence of univalent maps, still written {g n }, locally uniformly converging to a holomorphic map g on U 0 0 . By (4.2) the map g is not a constant. Thus g : We claim that the domain g(U 0 0 ) is contained in the Fatou set. Otherwise, for any domain V with V ⊂ g(U 0 0 ) and V ∩ J f = ∅, the iteration f n (V ) will eventually cover the sphere except two points. This contradicts the fact that f n (V ) ⊂ U 0 0 for sufficiently large n. Let W be a domain with cl(W ) ⊂ g(U 0 0 ). Note that each periodic Fatou component are supper attracting (since f is postcritically finite), then the claim above implies that f n (W ) converges to an attracting periodic orbit. On the other hand, as n is large enough, the map f n | W = g −1 n | W uniformly converges to the univalent map g −1 | W . It is a contradiction.
Proposition 4.6. The Thurston map F defined in (4.1) is expanding.
Proof. To show the expansion of F , we only have to prove that the maximum of diam σ (X n ) with X n ∈ X n (C F ) converges to 0 as n → 0. Note that π( X n \ K n ) = π( X n k ) for all X n ∈ X n (C f ). Then the proposition follows immediately from Lemmas 4.4 and 4.5.

Proof of the main theorem
Proof of Theorem 1.1. The proof follows the outline given in Section 4.1.
We first prove the sufficiency. Let f be a postcritically finite rational map with Sierpiński carpet Julia set. From Section 4.2 we obtain a semi-conjugacy π from the rational map f to an expanding Thurston map F , and two Jordan curves C f and C F .
We label post(f ) in C f by x 1 , · · · , x m , x m+1 = x 1 successively in the cyclic order, and denote by C f ( x i , x i+1 ) the arc on C f with endpoints x i and x i+1 . Set x i = π( x i ) for all i ∈ {1, . . . , m+1} and similarly define C F (x i , x i+1 ). It is clear that C F (x i , x i+1 ) = π(C f ( x i , x i+1 )). Moreover, by Lemma 4.4 there is an one to one correspondence between X n (C f ) and X n (C F ), characterized by the map X n (C f ) ∋ X n k → X n k := π( X n k ) ∈ X n (C F ) for each n ≥ 0. Let ψ : C f → C F be an orientation preserving homeomorphism such that ψ( x i ) = x i and ψ(C f ( x i , x i+1 )) = C F (x i , x i+1 ) for all i ∈ {1, . . . , m}. There exists then a homotopy h 0 : C f ×I → C F rel. post(f ) from ψ to π| C f by Lemma 3.2 (1). We extend ψ to an orientation preserving homeomorphism of C, also denoted by ψ, with ψ( X 0 k ) = X 0 k , k = 0, 1. It follows from Lemma 3.2 (2) that the homotopy h 0 can be extended to a homotopy H 0 : S 2 × I → S 2 rel. post(f ) from ψ to π. By the property of π that π −1 (x) is either a point in J f or the closure of a Fatou component, and the specific construction of homotopies in Lemma 3.2, we have that for each x i ∈ post(F ). We know that the Riemann sphere C and the 2-sphere S 2 admit a partition by the 1-tiles relative to (f, C f ) and (F, C F ) respectively, and the numbers of X 1 (C f ) and X 1 (C F ) are both 2d. For each j ∈ {1, . . . , 2d}, we define a map It is a composition of 3 onto homemorphisms, and hence a homeomorphism. It also satisfies that φ j (∂ X 1 j ) = ∂X 1 j and φ j | f −1 (post(f )) = π| f −1 (post(f )) . Using Lemma 3.2 again, we get a homotopy H 1 j : We define the map φ : C → S 2 by φ(z) := φ j (z) if z ∈ X 1 j , and the map H 1 : 1]. It is clear that φ is a homeomorphism and H 1 is a homotopy rel. f −1 (post(f )) from π to φ. With the same reason, the homotopy H 1 satisfies the similar property as H 0 , that is, (H 1 t ) −1 (p) = p, ∀t ∈ [0, 1) and (H 1 1 ) −1 (p) = [ p] for all p ∈ F −1 (post(F )), where p ∈ f −1 (post(f )) satisfy that π( p) = p.
Concatenating the homotopies H 0 , H 1 together, we get a homotopy H : C × I → S 2 rel. post(f ) from ψ to φ defined by The homotopy H satisfies that for each x i ∈ post(F ). By Lemma 3.4, the homeomorphisms ψ and φ are isotopic rel. post(f ).
We now turn to the necessity. Let f be a postcritically finite rational map. By Whyburn's characterization (see Section 2.4) and Lemma 2.2, in order to show that J f is a Sierpiński carpet, we just need to prove that the closures of any two distinct Fatou components are disjoint and each Fatou component is a Jordan domain.
Let f be Thurston equivalent to an expanding Thurston map F via h 0 , h 1 . Using isotopy lifting theorem (see [1,Proposition 11.1]) repeatedly, we obtain a sequence of homeomorphisms {h n } n≥0 such that h n • f = F • h n+1 and h n is isotopic to h n+1 rel. f −n (post(f )), i.e., the following diagram commutes.
Since F is expanding, by [1,Lemma 11.3], the sequence of homeomorphisms {h n } n≥0 uniformly converges to a continuous map h with respect to a metric ω on S 2 . We then get a semi-conjugacy h from f to F , i.e., F • h = h • f on C. Besides, the restriction is bijective. Because for all k ≥ n ≥ 0, h k = h n : f −n (post(f )) → F −n (post(F )) is bijective.
We claim that h(U ) is a singleton for each Fatou component U . By Sullivan's non-wandering Fatou component Theorem, we can assume that U is fixed by f . Let r ′ be a periodic internal ray of U with period q, and denote the center of U by c. For each n ≥ 0, we set r n := h qn (r ′ ), which is a lift of r 0 by F qn following the commutative diagram (4.3). By the expansion of F , it is proved in [1,Lemma 8.7] that diam ω (r n ) ≤ CΛ −qn → 0 as n → ∞, where Λ ≥ 1 and C are constants. So h(r ′ ) = h(c). As the periodic internal rays are dense in U , we get that h(U ) = h(c). By this claim and (4.4), the closures of distinct Fatou components are disjoint.
We are left to show that each Fatou component is a Jordan domain. Without loss of generality, let U be a fixed Fatou component. We assume that U is not a Jordan domain and argue by contradiction. Note that ∂U is locally connected, from the Böttcher's theorem there exist two internal rays of U landing at a common point in ∂U . The closure of their union is a Jordan curve bounding two domains W 0 and W 1 with W i ∩ J f = ∅, i ∈ {0, 1}.
We claim that each of the domains W 0 , W 1 contains a Fatou component. Otherwise, there is i ∈ {0, 1} such that f n (W i ) ⊆ (U ∪ J f ) for all n ≥ 0. By the topological transitivity of the Julia set, the set f n (W i ) for sufficiently large n, hence U ∪ J f , covers C except at most two points (see [16,Theorem 4.10]). It means that f has only one Fatou component U , impossible.
Let U 0 and U 1 be the Fatou components contained in W 0 and W 1 respectively. By the discussion above, we have that the images h(U ), h(U 0 ) and h(U 1 ) are pairwise different points. Consequently, the set h −1 (h(U )) contains the Jordan curve ∂W 0 = ∂W 1 ⊂ U , and is disjoint with U 0 , U 1 . It implies that S 2 \ h −1 (h(U )) is not connected. On the other hand, note that h is the limit of a sequence of homeomorphisms of S 2 . By [5, Lemma 3.1], such a map h has a property that S 2 \ h −1 (x) is connected for any x ∈ S 2 . It contradicts that S 2 \ h −1 (h(U )) is not connected. The proof of the necessity is completed.

Appendix
Proof of Lemma 3.4. Let H : S × I → S rel. ∂S ∪ P be a homotopy from id S to h. Set K := ∪ p∈P K p . See Lemma 3.4 for the definition of K p . We can choose a closed topological disk D p ⊂ S for each p ∈ P such that We claim that it is enough to prove the lemma in the case that h| γp = id γp for all p ∈ P . To see this, note first that the homotopy H induces a homotopy H| γp×I : γ p × I → S \ P from γ p to h(γ p ) for each p ∈ P . By [9, Theorem 2.1], there exists an isotopy φ : (S \ P ) × I → S \ P rel. ∂(S \ P ) such that φ 0 = id S\P and φ 1 | γp = h| γp for all p ∈ P . It can be also viewed as an isotopy φ : S × I → S rel. ∂S ∪ P . We then get a homotopy Φ : We now assume that h| γp = id γp for all p ∈ P . We claim that if each γ p is fixed by the homotopy H, i.e., H(x, t) = x for all t ∈ [0, 1], x ∈ γ p , then the conclusion holds. Let the homotopy H satisfy the property of this claim. Applying (2) in Lemma 3.2 to h| Dp and id Dp with A = γ p , we have that h| Dp is isotopic to id Dp rel. γ p for each p ∈ P . Set M := S \ (∪ p∈P int(D p )). For each p ∈ P , we construct a radical projection π p : D p \ {p} → γ p z → α −1 p (α p (z)/|α p (z)|), where α p : D p → cl(D) is a homeomorphism with α p (p) = 0. Note that for any x ∈ M , the curve H(x, t), t ∈ I avoids P . Then we define a map H : M × I → S by H(x, t) := π p • H(x, t), if H(x, t) ∈ D p for some p ∈ P ; H(x, t), otherwise.
Thus we get a homotopy H| M ×I : M × I → M rel. ∂M . By Theorem 3.3, the maps id M and h| M are isotopic rel. ∂S. From the argument above, we see that id S is isotopic to h rel. ∂S ∪ P .
Consequently, we remain to find a homotopy H : S × I → S rel. ∂S ∪ P ∪ (∪ p∈P γ p ) from id S to h under the assumption that h| γp = id γp for all p ∈ P .
Cutting the disks D p , p ∈ P from S, we get the surfaces D p , p ∈ P and M following the notations above. Note that each γ p belongs to both D p and M . For distinguishing, we denote the γ p in D p by γ − p and that in M by γ + p . And a point ξ ∈ γ p is represented by ξ ± in γ ± p . We then paste each D p with M by the annulus A p := γ p × [−1, 1]. Precisely, let ≈ be an equivalence relation on the disjoint union (⊔ p∈P D p ) ⊔ (⊔ p∈P A p ) ⊔ M such that x ≈ y if and only if x = y, or x = ξ ± ∈ γ ± p and y = ξ × {±1} ∈ A p for some ξ ∈ γ p and p ∈ P . The quotient space is clearly homeomorphic to S.
For a point x ∈ S b , it has a natural name x as in the original surface S if x ∈ (⊔ p∈P D p ) ⊔ M ; and it is parameterized by {(ξ, s) : ξ ∈ γ p , s ∈ [−1, 1]} if x ∈ A p for some p ∈ P . With these notations, we define a map H b : S b × I → S by H(ξ, t|s|), if x = (ξ, s) ∈ A p for some p ∈ P .
Then the map H b : S b × I → S is a homotopy rel. ∂S ∪ P ∪ (∪ p∈P γ p × {0}).
As S b is homeomorphic to S, we identify S b with S and identify a point ξ × 0 ∈ γ p × 0 ⊂ A p with ξ ∈ γ p ⊂ S for p ∈ P . In such a view, each A p is an annulus neighborhood of γ p , and H b : S × I → S is a homotopy relative to ∂S ∪ P ∪ (∪ p∈P γ p ).
By the definition of H b , the map H b | S×0 (resp. H b | S×1 ) is homotopic to id S (resp. h) rel. ∂S ∪ P ∪ (∪ p∈P γ p ). It follows that Then the proof is completed.