Weak Tangents of Metric Spheres

Transcription

1 Weak Tangents of Metric Spheres Angela Wu University of California, Los Angeles New Developments in Complex Analysis and Function Theory, 2nd July, 2018 Angela Wu Weak Tangents of Metric Spheres

2 Weak Tangent: Definition Let (X, d) be a metric space. What is a weak tangent of (X, d)? 2 / 11

3 Weak Tangent: Definition Let (X, d) be a metric space. What is a weak tangent of (X, d)? Gist: x n X, λ n +, (X, x n, λ n d) (T, x, d T ) 2 / 11

4 Weak Tangent: Definition Let (X, d) be a metric space. What is a weak tangent of (X, d)? Gist: x n X, λ n +, More precisely: R > 0, (X, x n, λ n d) (T, x, d T ) B (X,λnd)(x n, R) B (T,dT )(x, R). 2 / 11

5 Weak Tangent: Definition Let (X, d) be a metric space. What is a weak tangent of (X, d)? Gist: x n X, λ n +, More precisely: R > 0, (X, x n, λ n d) (T, x, d T ) B (X,λnd)(x n, R) B (T,dT )(x, R). Absolute rigor: R > 0, ε > 0, N N, n N, d GH (B (X,λnd)(x n, R + ε), B (T,dT )(x, R)) < ε. 2 / 11

6 Weak Tangent: Definition Example If (X, d) is a compact Riemannian n-manifold, then every weak tangent of (X, d) is (R n, 0). Example The standard 1/3-Cantor set has uncountably many weak tangents. 3 / 11

7 Quasisymmetry Definition A quasisymmetry ϕ : X Y between two metric spaces (X, d X ) and (Y, d Y ) is a homeomorphism such that for all x, y, z X with x z, d Y (ϕ(x), ϕ(y)) d Y (ϕ(x), ϕ(z)) η ( ) dx (x, y). d X (x, z) where η : [0, ) [0, ) is a homeomorphism. Intuition: there exists C > 1 such that for every ball B X, there exists another ball B Y such that B ϕ(b) CB. f (B) 4 / 11

8 Quasisymmetry Definition A quasisymmetry ϕ : X Y between two metric spaces (X, d X ) and (Y, d Y ) is a homeomorphism such that for all x, y, z X with x z, d Y (ϕ(x), ϕ(y)) d Y (ϕ(x), ϕ(z)) η ( ) dx (x, y). d X (x, z) where η : [0, ) [0, ) is a homeomorphism. Remark: If ϕ : X Y is a quasisymmetry, then ϕ 1 : Y X is a quasisymmetry. 4 / 11

9 Weak Tangents and Quasisymmetries Lemma If ϕ : X Y is a quasisymmetry, and if (X, x n, λ n d X ) T, then for some µ n, (Y, ϕ(x n ), µ n d Y ) has a converging subsequence whose limit T is quasisymmetric to T. 5 / 11

10 Weak Tangents and Quasisymmetries Lemma If ϕ : X Y is a quasisymmetry, and if (X, x n, λ n d X ) T, then for some µ n, (Y, ϕ(x n ), µ n d Y ) has a converging subsequence whose limit T is quasisymmetric to T. Theorem (Kinneberg [3]) A doubling metric circle C is a quasicircle if and only if every weak tangent of C is quasisymmetric to (R, 0). 5 / 11

11 Weak Tangents and Quasisymmetries Lemma If ϕ : X Y is a quasisymmetry, and if (X, x n, λ n d X ) T, then for some µ n, (Y, ϕ(x n ), µ n d Y ) has a converging subsequence whose limit T is quasisymmetric to T. Theorem (Kinneberg [3]) A doubling metric circle C is a quasicircle if and only if every weak tangent of C is quasisymmetric to (R, 0). Theorem (W. [4]) For all n 2, there exists a doubling, LLC metric space X homeomorphic to S n such that every weak tangent of X is isometric to (R n, 0) but X is not quasisymmetric to S n. 5 / 11

12 Weak Tangents and Quasisymmetries Theorem (Bonk, Kleiner [1]) Suppose Z is a uniformly perfect, doubling compact metric space, and G Z is a uniformly quasi-möbius action for which the induced action G Tri(Z) is cocompact. If (T, p) is a weak tangent of Z, then there exists a quasi-möbius homeomorphism h : (Ŝ, d p ) Z. 6 / 11

13 Weak Tangents and Quasisymmetries Theorem (Bonk, Kleiner [1]) Suppose Z is a uniformly perfect, doubling compact metric space, and G Z is a uniformly quasi-möbius action for which the induced action G Tri(Z) is cocompact. If (T, p) is a weak tangent of Z, then there exists a quasi-möbius homeomorphism h : (Ŝ, d p ) Z. Taking G = finitely generated infinite hyperbolic group, Z = G: Corollary Suppose G is homeomorphic to S n. If any weak tangent of G is quasisymmetric to (R n, 0), then G is quasisymmetric to S n. 6 / 11

14 Weak Tangents and Quasisymmetries Theorem (Bonk, Kleiner [1]) Suppose Z is a uniformly perfect, doubling compact metric space, and G Z is a uniformly quasi-möbius action for which the induced action G Tri(Z) is cocompact. If (T, p) is a weak tangent of Z, then there exists a quasi-möbius homeomorphism h : (Ŝ, d p ) Z. Taking G = finitely generated infinite hyperbolic group, Z = G: Corollary Suppose G is homeomorphic to S n. If any weak tangent of G is quasisymmetric to (R n, 0), then G is quasisymmetric to S n. Conjecture (Cannon s conjecture) If G is homeomorphic to S 2, then G is quasisymmetric to S 2. 6 / 11

15 Weak Tangents and Expanding Thurston Maps f : S 2 S 2 is an expanding Thurston map i.e 7 / 11

16 Weak Tangents and Expanding Thurston Maps f : S 2 S 2 is an expanding Thurston map i.e 1 branched covering : x S 2, open U x, homeo ϕ, ψ, (U, x) f (f (U), f (x)) ϕ ψ z z d (D, 0) (D, 0). 7 / 11

17 Weak Tangents and Expanding Thurston Maps f : S 2 S 2 is an expanding Thurston map i.e 1 branched covering 2 postcritically finite: {f n (c) : n N, c critical point} < 7 / 11

18 Weak Tangents and Expanding Thurston Maps f : S 2 S 2 is an expanding Thurston map i.e 1 branched covering 2 postcritically finite: {f n (c) : n N, c critical point} < 3 for every x S 2 there exists open x U S 2, lim sup{diam V : V connected component of f n (U)} = 0. n 7 / 11

19 Weak Tangents and Expanding Thurston Maps f : S 2 S 2 is an expanding Thurston map i.e 1 branched covering 2 postcritically finite: {f n (c) : n N, c critical point} < 3 for every x S 2 there exists open x U S 2, lim sup{diam V : V connected component of f n (U)} = 0. n Example (The 2 2-subdivision map) a d b c a d f D C a b a A B 7 / 11

20 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles 8 / 11

21 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles For general expanding Thurston maps: 8 / 11

22 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles For general expanding Thurston maps: 1 There exists N N and f N -invariant Jordan curve C containing all postcritical points [2]. WLOG N = 1. 8 / 11

23 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles For general expanding Thurston maps: 1 There exists N N and f N -invariant Jordan curve C containing all postcritical points [2]. WLOG N = 1. 2 C gives us 2 0-tiles. 8 / 11

24 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles For general expanding Thurston maps: 1 There exists N N and f N -invariant Jordan curve C containing all postcritical points [2]. WLOG N = 1. 2 C gives us 2 0-tiles. 3 f 1 (C) divides 0-tiles into 1-tiles. 8 / 11

25 Visual Spheres of Expanding Thurston Maps f f 2-tiles 1-tiles 0-tiles For general expanding Thurston maps: 1 There exists N N and f N -invariant Jordan curve C containing all postcritical points [2]. WLOG N = 1. 2 C gives us 2 0-tiles. 3 f 1 (C) divides 0-tiles into 1-tiles. 4 f n (C) gives us n-tiles. 8 / 11

26 Visual Spheres of Expanding Thurston Maps Proposition (Bonk and Meyer [2]) There exists Λ > 1 and a metric ρ on S 2 that generates the topology of S 2 and such that diam ρ (n-tile) Λ n. ρ is a visual metric with respect to f Λ is the expansion factor of ρ. (S 2, ρ) is a visual sphere. visual metric always exists for f but not unique. two visual spheres for f are snowflake equivalent, therefore quasisymmetric. 9 / 11

27 Visual Spheres of Expanding Thurston Maps Theorem Let f be an expanding Thurston map with no periodic critical point and ρ be a visual metric with respect to f. TFAE: (i) f is Thurston equivalent to a rational map. (ii) (S 2, ρ) is a quasisphere. (iii) Every weak tangent of (S 2, ρ) is quasisymmetric to (R 2, 0). (iv) Some weak tangent of (S 2, ρ) is quasisymmetric to (R 2, 0). (i) (ii) Bonk and Meyer [2]. (ii) = (iii) Lemma. (iii) = (iv) Clear since (S 2, ρ) has a weak tangent. (iv) = (ii) W. 10 / 11

28 Bibliography I M. Bonk and B. Kleiner Rigidity for quasi-möbius group actions, J. Differential Geom., 61(1):81-106, M. Bonk and D. Meyer Expanding Thurston maps, American Mathematical Soc., K. Kinneberg, Conformal dimension and boundaries of planar domains, Trans. Amer. Math. Soc. 369(9) , A. Wu, A metric sphere not a quasisphere but for which every weak tangent is Euclidean, arxiv: , / 11