Groupoid actions on Fractafold Bundles

Transcription

1 Groupoid actions on Fractafold Bundles Marius Ionescu 1 Alex Kumjian 2 1 Colgate University 2 University of Nevada, Reno Conference on Noncommutative Geometry & Quantum Groups in honor of Marc A. Rieffel Fields Institute, 27 June 2013

2 Introduction Our goal is to find and analyze symmetries of a fractal associated to an iterated function system, F 1,..., F n. Stricharz constructed a family of fractafold blowups of the invariant set of an iterated function system parameterized by X = {1,..., n} N. He observed that two such blowups are naturally homeomorphic if the parametrizing sequences are eventually the same. We construct a bundle L of fractafold blowups endowed with the inductive limit topology over the sequence space X. The symmetries observed by Stricharz give rise to a groupoid action on the bundle L. The resulting action groupoid is isomorphic to a Renault-Deaconu groupoid determined by a local homeomorphism on L. The action groupoid is essentially free and has a dense orbit. This is work in progress.

3 Renault-Deaconu Groupoid Let σ : X X be a local homeomorphism. Define ([R1], [D], [R2]): G(X, σ) := {(x, l m, y) : σ l x = σ m y} X Z X. Then G(X, σ) is an étale groupoid with G(X, σ) 0 = X and with the following structure maps: i(x) = (x, 0, x) s(x, n, y) = y r(x, n, y) = x (x, l, y)(y, m, z) = (x, l + m, z) (x, n, y) 1 = (y, n, x) G(X, σ) is called the Renault-Deaconu groupoid associated to σ. Example: Let X = {1,..., n} N and let σ be the shift, σ(x) i = x i+1. The corresponding Renault-Deaconu groupoid is often called the Cuntz groupoid (since C (X, σ) = O n ).

4 Groupoid actions Let G be a Hausdorff groupoid with unit space G 0 = X, let ρ : Z X be a continuous open surjection. G acts on Z if there is a continuous map δ : G Z Z such that i. ρ(γ z) = r(γ) ii. γ 1 (γ 2 z) = (γ 1 γ 2 ) z iii. i ρ(z) z = z δ(γ, z) = γ z Note G δ Z = G Z may itself be regarded as a groupoid, called the action groupoid, with topology inherited from G Z, (G δ Z) 0 = Z and the following structure maps: i(z) = (i ρ(z), z) s(γ, z) = z r(γ, z) = γ z (γ 1 γ 2, z) = (γ 1, γ 2 z)(γ 2, z) (γ, z) 1 = (γ 1, γ z)

5 Actions by Renault-Deaconu Groupoids If the action is that of a Renault-Deaconu groupoid, the resulting action groupoid is itself of the same form. Theorem Let σ : X X be a local homeomorphism and let δ : G(X, σ) Z Z be an action of G(X, σ) on Z (with ρ : Z X as above). Then the map σ : Z Z defined by σ(z) = (σ(ρ(z)), 1, ρ(z)) z is a local homeomorphism. Moreover, ρ σ = σ ρ and G(Z, σ) = G(X, σ) δ Z. Sketch of proof: The isomorphism is given by (w, n, z) ((ρ(w), n, ρ(z)), z) with inverse given by ((x, n, y), z) ((x, n, y) z, n, z).

6 Iterated Function Systems We consider an iterated function system (IFS) of the following type. Fix n > 1 and let (Y, d) be a complete metric space. For 1 i n, let F i : Y Y be functions for which there are constants 0 < r i R i < 1, such that for all x, y Y, i = 1,..., n. r i d(x, y) d(f i (x), F i (y)) R i d(x, y), There is a unique nonempty compact invariant set K (see [H]): K = F 1 (K ) F n (K ) Endow X = {1,..., n} N with the usual topology generated by cylinder sets Z(α) (where α is a finite word). There is a continuous map p : X K such that for x X, y Y p(x) = lim k F x1 F xk (y)

7 Fractafold Blowups For x X, Stricharz (see [S]) considered the fractafold blowup L x = m F 1 x 1 F 1 x m (K ) He observed that L x = Ly if x and y have the same tail. For α = m, set L(α) = Fα 1 1 Fα 1 m (K ). For x X, endow L x with the inductive limit topology L x = m L(x 1,..., x m ) and assemble them into a bundle over X. Set L m := α =m Z(α) L(α) X Y, L := m Endow L with the inductive limit topology. Note L m is compact and L m L m+1. The canonical map π : L X is a continuous open surjection and L x = π 1 ({x}) L m

8 The Groupoid Action on the Fractafold Bundle Let X = {1,..., n} N, define σ(x) i = x i+1 and set G = G(X, σ). We define an action of G on L as follows: For γ = (x, m, y) and t L y, there are z X and finite words α and β such that x = αz, y = βz, m = α β, t L(β). Then setting j = α and k = β, δ is given by δ(γ, t) = γ t = F 1 α 1 F 1 α j F βk F β1 (t) It is routine to show that the action δ is well defined and continuous. For x X and 1 i n, F i extends to a homeo. F i : L ix L x. We define a local homeomorphism σ : L L by σ(x, t) = (σ(x), F x1 (t)) = (σ(x), 1, x) (x, t) Hence, as in the theorem, we have G(Z, σ) = G(X, σ) δ Z.

9 An Example Fix n > 1, r (0, 1) and let {e 1,..., e n } be the standard basis in R n. For 1 j n, define F j : R n R n by F j (t) = rt + (1 r)e j, t R n. Then F 1,..., F n form an IFS on R n. The invariant set K consists of all sums of the form 1 r r r k e xk k=1 where x X = {1,..., n} N If n = 3 and r = 1/2, K is the Sierpinski Triangle. Note that K is totally disconnected if r < 1/2. For α {1,..., n} m, L(α) consists of all points of the form 1 r m r m+1 r j e xj r i e αi j=1 i=1

10 Some references. [D] Deaconu, Groupoids associated with endomorphisms. [DKM] Deaconu, Kumjian and Muhly, Cohomology of topological graphs and Cuntz-Pimsner algebras. [H] Hutchinson, Fractals and self-similarity. [R1] Renault, A groupoid approach to C*-algebras. [R2] Renault, Cuntz-like algebras. [S] Stricharz, Fractals in the large.