Some invariant properties of quasi-M\"obius maps

We investigate properties which remain invariant under the action of quasi-M\"obius maps of quasi-metric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)M\"obius spaces. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)M\"obius spaces as well.


Introduction
Let (X, d) be a metric space. X is doubling if there exists a constant D > 0, such that every ball of finite radius can be covered by at most D balls of half the radius. X is uniformly disconnected if there exists a constant θ < 1, such that X contains no θ-chain, i.e. a sequence of (at least 3 distinct) points (x 0 , x 1 , . . . , x n ) such that d(x i , x i+1 ) ≤ θd(x 0 , x n ). holds. Here the cross-ratio cr is given by The aim of this paper is to prove the following two theorems: Theorem 2 (Invariance of uniform disconnectedness under quasi-Möbius maps). Let (X, d) be a metric uniformly disconnected space and let f : The results are related to results of Lang-Schlichenmaier [5] and Xie [11] who proved that quasi-symmetric maps respectively quasi-Möbius maps preserve the Nagata dimension of metric spaces. The present work has been inspired by the article of Xie [11] and the work of Väisälä [10]. We note that a space is doubling if and only if it has finite Assouad dimension [7]. However the Assouad dimension is not a quasi-symmetric (and therefore also not a quasi-Möbius) invariant [9].
We would like to note that we have been informed that Theorem 1 is also a consequence of a published result of Li-Shanmugalingam [6].
It is well known that uniform disconnectedness is invariant under quasisymmetric maps [7,4]. However its behaviour under quasi-Möbius maps has not been studied before.
The related property of uniform perfectness has been shown to be invariant under the metric inversion in [8]. It is therefore also invariant under quasi-Möbius maps.
In Appendix A we prove a slight generalization of Theorem 1 and Theorem 2 for K-quasi-metric spaces.
If Ω(d) is non empty we call ω ∈ Ω(d) the infinitely remote point of X. By abuse of notation we may write ∞ for the point ω.

Doubling Property
We call a metric space doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius.

Uniform Disconnectedness
For θ < 1 we call a sequence of (at least 3 distinct) points ( holds for all i ∈ {0, 1, . . . , n − 1}. A metric space is called uniformly disconnected with constant θ if it contains no θ-chains. 1

Quasi-Möbius and Quasi-Symmetric Maps
We call a homeomorphism f : .
3 Invariance of Doubling Property

Preparations for the Proof
For the proof we need the following proposition of Xie and a result of Väisälä which we cite verbatim Proposition 1 (Proposition 3.6 in [11]). Let f : (X 1 , d 1 ) → (X 2 , d 2 ) be a quasi-Möbius homeomorphism. Then f can be written as is either a metric inversion or the identity map on the metric space (X i , d i ).
Proposition 2 (Theorem 3.10 in [10]). Let (X, d) be an unbounded metric space and let f : Remark 1. Let (X, d) be an unbounded space. Then we can build the completed space with respect to the infinitely remote pointX := X ∪{∞} together with an extended metricd.
Theorem 3. Let (X, d) be an metric doubling space with doubling constant D, where d is an extended metric [3] and denote by ∞ ∈ X the infinitely remote point in (X, d). Furthermore let p ∈ X with p = ∞ and let i p be given by . We therefore only need to show the theorem for unbounded X.
We have the following relation for all x, y ∈ X \ {p} [2]: .
≤ r} be the ball of radius r in the space (X, d p ). We consider the following two cases . Take y 0 ∈ A ′ . For any two points x, y ∈ A ′ we have by definition of the metric d p and the above relation that In particular we know that A ′ ⊆ B 32 and therefore also The space (X, d) is doubling and we can find D N balls b i of radius 32 we have for all x, y ∈b i : In particular for N := 10 we know that we have constructed a cover of B ′ ⊆ A ′ ∪ B ′ 1 2 r (∞) by D 10 + 1 balls of radius 1 2 r.

In case that
We therefore have B ′ ⊆ B 4r dp(∞,B ′ ) 2 (x 0 ) and by the doubling property of (X, d) we can cover by D N balls b i of radius 4r dp(∞,B ′ ) 2 2 −N with center points x i . Letb i := b i ∩ B ′ , then we have for any two x, y ∈b i : Furthermore we have In conclusion we get that It therefore follows that if we take N := 9, then we have a covering of B ′ by D 9 balls of radius 1 2 r. Case 2 follows directly from Theorem 3. In the situation of 1, d ′ is a metric inversion d p where p is an isolated point in X. That is there exists a ǫ > 0 such that d(p, x) > ǫ for all x ∈ X \ {p}. The proof of Theorem 3 still holds.

Invariance of Uniform Disconnectedness
The proof of Theorem 2 will again make use of some of the propositions from the previous sections.
In the following let (X, d) be a metric space, p ∈ X and θ ≤ 1 32 . We assume that (X, d p ) is not θ-uniformly disconnected, in particular there is some θ-chain (x 0 , x 1 , . . . , x n ) in (X \ {p}, d p ). We keep this notation for the rest of this section. In addition we introduce the following notation for convenience: Let r i := d(p, x i ), l := d(x 0 , x n ) and l i := d(x i , x i+1 ). This is illustrated in Figure 1. Without loss of generality we can assume r n ≥ r 0 .

Remark 3. The condition for
On the other hand if .
Proof. Assume for a contradiction that r s 3 √ 4θ < r 0 and r s+1 3 √ 4θ < r 0 . Then from the condition in the remark above it follows which is a contradiction.
Proof. By the previous lemma we know that there must be some index q such that r q 3 √ 4θ ≥ r 0 and for all i ∈ {0, . . . , q − 1} we have that r i 3 √ 4θ < r 0 . We claim that (x q , x q−1 , . . . , x 1 , x 0 , p) is a 3 √ 4θ-chain in (X, d). If this were not so, there would be some i ∈ {0, . . . , q − 1} for which r q which is a contradiction to the triangle inequality of the metric space (X, d).
Proof of Theorem 2. The proof of the theorem now follows directly from Proposition 1. l s x n x 0 x 1 x 2 x q x s

Applications of the Theorems
For the following we need a short definition [4]: Let F be a finite set with k ≥ 2 elements and let F ∞ denote the set of sequences In particular we have L(x, x) = ∞ and L(x, y) = 0 if x 1 = y 1 . Given 0 < a < 1 set ρ a (x, y) = a L(x,y) . This defines an ultrametric on F ∞ . We call (F ∞ , ρ a ) the symbolic k-Cantor set with parameter a.
As an application of the theorems we provide a generalization of the following result by David and Semmes: Proposition 4 (Proposition 15.11 (Uniformization) in [4]). Suppose that (M, d) is a compact metric space which is bounded, complete, doubling, uniformly disconnected, and uniformly perfect. Then M is quasi-symmetrically equivalent to the symbolic Cantor set F ∞ , where we take F = {0, 1} and we use the metric ρ a on F ∞ with parameter a = 1 2 .
We can generalize this result as follows: .
Then the space (M,d p ) is bounded and satisfies all the properties of the above proposition: The map f : (X, d) → (X,d p ) given by d →d p is Möbius. By Theorem 2 and Theorem 1, doubling and uniformly disconnectedness are invariant under Möbius maps. The invariance of uniformly perfectness follows from [8], and the invariance of completeness follows from [1]. Totally boundedness follows from the doubling property and therefore the space (X,d p ) is compact.
We can apply the same idea to Proposition 16.9 in [4] and we get: This follows from the above remarks and the invariance of Ahlfors regularity under d →d p as shown in [6].
Furthermore we have for all x ∈ A ′ that d λ (∞, x) = L λ(x) > 1 2 r and therefore also λ(x) < 2L r . Combining both equations we get that for all x, y ∈ A ′ we have Without loss of generality assume x 0 ∈ A ′ . By the doubling property of (X, d) we can cover B K ′2 4L 2 r (x 0 ) by at most D N balls b i of radius By the assumption there is ax ∈ B ′ ∩ B ′ 1 2 r (∞) and we have for x ∈ B ′ that d λ (x,x) ≤ K ′2 r, therefore we also have L λ(x) = d λ (x, ∞) ≤ K ′4 r and λ(x) ≥ L K ′4 r . In conclusion we get for all x, y ∈b i : In particular for N := ⌈log 2 (8K ′10 K)⌉ we get a cover of B ′ by at most D N + 1 balls of half the radius. By the doubling property of (X, d) we can find D N balls b i of radius rL 2 d λ (B ′ ,∞) 2 2 −N covering B ′ . Letb i := b i ∩ B ′ , then we have for any x, y ∈b i : Furthermore for any x ∈ B ′ we have We can combine the estimates to get In particular for N := ⌈log 2 (8KK ′4 )⌉ we have constructed a covering by D N balls of radius at most 1 2 r.