Quasisymmetric uniformization - PDF Free Download

Quasisymmetric uniformization Daniel Meyer Jacobs University May 1, 2013

Quasisymmetry X, Y metric spaces, ϕ: X Y is quasisymmetric, if ( ) ϕ(x) ϕ(y) x y ϕ(x) ϕ(z) η, x z for all x, y, z X, η : [0, ) [0, ) homeo.

Quasisymmetry X, Y metric spaces, ϕ: X Y is quasisymmetric, if ( ) ϕ(x) ϕ(y) x y ϕ(x) ϕ(z) η, x z for all x, y, z X, η : [0, ) [0, ) homeo. z y ϕ ϕ(y) ϕ(z) x ϕ(x)

Quasisymmetry X, Y metric spaces, ϕ: X Y is quasisymmetric, if ( ) ϕ(x) ϕ(y) x y ϕ(x) ϕ(z) η, x z for all x, y, z X, η : [0, ) [0, ) homeo. z y ϕ ϕ(y) ϕ(z) x ϕ(x) qs qc qs qc in R n.

Quasisymmetry Quasisymmetry is global notion of quasiconformality. Right notion for boundary at infinity of Gromov-hyperbolic spaces equipped with visual metric. Rough isometry on a space changes visual boundary by quasisymmetry. Quasisymmetric uniformization: when is a metric space X qs equivalent to some standard space S.

Quasisymmetric uniformization Qs uniformization problem: given metric space X, is it qs to some standard metric space S?

Quasisymmetric uniformization Qs uniformization problem: given metric space X, is it qs to some standard metric space S? Theorem (Tukia-Väisälä 80) J Jordan curve, J quasicircle (: J = f (S 1 ), f : S 1 J qs ) J bounded turning and doubling. J is bounded turning, if for all x, y J diam J[x, y] K x y, here J[x, y] is the (smaller) arc between x, y, and K < constant.

Snowflake f Have f (x) f (y) x y α, α = log 3/ log 4. f snowflake equivalence. (S, d) metric space, d α, 0 < α < 1, then (S, d α ) snowflaked space.

Snowflake f Have f (x) f (y) x y α, α = log 3/ log 4. f snowflake equivalence. (S, d) metric space, d α, 0 < α < 1, then (S, d α ) snowflaked space. Theorem (M,2010) J weak-quasicircle J bounded turning.

Catalogs of Quasicircles Rohde-snowflake. Fix 1/4 < p < 1/2. Start with square, in each step divide line segment into four, or replace by generator. In the limit obtain Rohde-snowflake R R p. p p p p 1/4 1/4 1/4 1/4 Theorem (Rohde, 2001) J C Jordan curve. Then J quasicircle J bi-lipschitz to some R R p, there exists bi-lipschitz f : C C s.t. f (R) = J.

Catalog of Quasicircles Theorem (Herron-M., 2012) There is a simple catalog of Jordan curves that contains all quasicircles up to bi-lipschitz equivalence. Fix 1/2 < p < 1 p p 1/2 1/2 Works in slightly modified form for bounded turning curves (not doubling).

Qs uniformization S qs to X. X = S 1 well understood. X Sierpiński carpet.

Qs uniformization S qs to X. X = S 1 well understood. X Sierpiński carpet. Theorem (Bonk-Merenkov, 2013) Distinct carpets are not qs, have only obvious qs-symmetries.

Qs uniformization S qs to X. X = S 1 well understood. X Sierpiński carpet. Theorem (Bonk-Merenkov, 2013) Distinct carpets are not qs, have only obvious qs-symmetries. Theorem (Bonk, 2011) S Ĉ carpet, boundaries are uniform quasicircles, uniformly relatively separated. Then S qs to round carpet.

Quasispheres When is a metric sphere X qs to S 2 (a quasisphere)?

Quasispheres When is a metric sphere X qs to S 2 (a quasisphere)? Conjecture (Cannon I) G is a Gromov hyperbolic group with G homeo to S 2. Then G H 3 isometric, properly discontinuous, and cocompact. Conjecture (Cannon II) G is a Gromov hyperbolic group with G homeo to S 2. Then G is a quasisphere.

Snowballs Examples of quasispheres: [David-Toro, 99] S R 3 that admit parametrization f : R 2 S R 3 s.t. α = 1 ɛ. f (x) f (y) x y α,

Snowballs Examples of quasispheres: [David-Toro, 99] S R 3 that admit parametrization f : R 2 S R 3 s.t. α = 1 ɛ. f (x) f (y) x y α, Alexander s horned sphere is a quasisphere.

Snowballs Examples of quasispheres: [David-Toro, 99] S R 3 that admit parametrization f : R 2 S R 3 s.t. α = 1 ɛ. f (x) f (y) x y α, Alexander s horned sphere is a quasisphere. Snowball S, fractal surface constructed as the snowflake.

Limit surface S is called a snowball. Rectifiably connected, for all x, y S there is a rectifiable curve γ xy S (from x to y), with length(γ xy ) x y.

Limit surface S is called a snowball. Rectifiably connected, for all x, y S there is a rectifiable curve γ xy S (from x to y), with length(γ xy ) x y. Quasisphere, map f : S S 2 may be extended qc to f : R 3 R 3.

Polyhedrons as Riemann surfaces Map cone to disk by z z α. Conformal except at cone point. z 2π/θ θ= cone angle

Polyhedrons as Riemann surfaces Map cone to disk by z z α. Conformal except at cone point. z 2π/θ θ= cone angle Given polyhedral use above as chart around vertex. Other charts: ϕ

Qs-uniformization of S Uniformize polyhedral approximations to find qs f : S S 2.

Fractal spheres generated by rational maps Consider slightly different snowball. Approximations are polyhedral spheres. Map from first to zero-th approximation h : S 1 S 0 is branched covering.

Fractal spheres generated by rational maps Consider slightly different snowball. Approximations are polyhedral spheres. Map from first to zero-th approximation h : S 1 S 0 is branched covering. Uniformize S 1, S 0. h S 1 S 0 ϕ 1 ϕ 2 Ĉ f Ĉ Note that h is holomorphic in flat charts. Thus f is holomorphic, i.e., rational. Can arrange that ϕ 0, ϕ 1 maps three vertices to the same points. The map f is postcritcally finite (each critical point has a finite orbit).

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3.

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 2

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 3

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 4

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 5

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 6

Rational maps and conformal tilings In fact: f (z) = 1 + ω 1 z 3, ω = e 4πi/3. 0 3:1 3:1 1 ω f 7 Uniformizations of approximations. Yields a qs-map of S to Ĉ.

Visual metric Reverse the previous process. Let f : Ĉ Ĉ be a rational map, postcritically finite, J(f ) = Ĉ.

Visual metric Reverse the previous process. Let f : Ĉ Ĉ be a rational map, postcritically finite, J(f ) = Ĉ. Let C Ĉ with post(f ) C.

Visual metric Reverse the previous process. Let f : Ĉ Ĉ be a rational map, postcritically finite, J(f ) = Ĉ. Let C Ĉ with post(f ) C. n-tile = closure of component of Ĉ \ f n (C).

Visual metric Reverse the previous process. Let f : Ĉ Ĉ be a rational map, postcritically finite, J(f ) = Ĉ. Let C Ĉ with post(f ) C. n-tile = closure of component of Ĉ \ f n (C). Want: metric in which each n-tile has roughly same diameter Λ n for some Λ > 1.

Visual metric A Gromov-product : m(x, y) := max{n : there ex. n-tiles X x, Y y, X Y }

Visual metric A Gromov-product : m(x, y) := max{n : there ex. n-tiles X x, Y y, X Y } A visual metric for f is a metric d on Ĉ satisfying d(x, y) Λ m(x,y) for all x, y Ĉ and a constant expansion factor Λ > 1. Not unique, but two visual metrics d 1, d 2 are snowflake equivalent. for constants C 1, α > 0. 1 C d 1(x, y) d 2 (x, y) α Cd 1 (x, y),

Expanding Thurston Maps Visual metrics can also be defined for expanding Thurston maps.

Expanding Thurston Maps Visual metrics can also be defined for expanding Thurston maps. Thurston map f : S 2 S 2 : branched cover of S 2, locally z z d, after homeomorphic coordinate changes, postcritically finite, expanding. This means J(f ) = Ĉ if f is rational. In general let C post(f ) be (any) Jordan curve. Then mesh S 2 \ f n (C) 0, as n. Independent of the chosen curve C.

Thurston s theorem Thurston gives characterization when a Thurston map is (equivalent to) a rational map [Douady-Hubbard 93].

Thurston s theorem Thurston gives characterization when a Thurston map is (equivalent to) a rational map [Douady-Hubbard 93]. This is the analog of Cannon s conjecture when does a map that acts topologically as a rational map is a rational map? There are (many) Thurston maps that are obtructed.

Existence of visual metrics Theorem (M. Bonk-D.M. 10, Haïssinsky-Pilgrim 09) There is combinatorial expansion factor Λ 0 = Λ 0 (f ) > 1 such that

Existence of visual metrics Theorem (M. Bonk-D.M. 10, Haïssinsky-Pilgrim 09) There is combinatorial expansion factor Λ 0 = Λ 0 (f ) > 1 such that Λ Λ 0 for each visual metric.

Existence of visual metrics Theorem (M. Bonk-D.M. 10, Haïssinsky-Pilgrim 09) There is combinatorial expansion factor Λ 0 = Λ 0 (f ) > 1 such that Λ Λ 0 for each visual metric. For each 1 < Λ < Λ 0 there is a visual metric d with expansion factor Λ. In fact d can be chosen as follows. For every x S 2 there exists neighborhood U x such that d(f (x), f (y)) d(x, y) = Λ, for all y U y.

Geometry of (S 2, d) The geometry of (S 2, d) encodes properties of f.

Geometry of (S 2, d) The geometry of (S 2, d) encodes properties of f. (S 2, d) is doubling f has no periodic critical points (Bonk-M. 10).

Geometry of (S 2, d) The geometry of (S 2, d) encodes properties of f. (S 2, d) is doubling f has no periodic critical points (Bonk-M. 10). (S 2, d) is snowflake equivalent to S 2 f is a Lattès map (M. 09). (S 2, d) is qs equivalent to S 2 f is rational (M. 02, 04, Bonk-Kleiner 05, Haïssinsky-Pilgrim 09, Bonk-M. 10).