Self-similar fractals as boundaries of networks

Transcription

1 Self-similar fractals as boundaries of networks Erin P. J. Pearse University of Oklahoma 4th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals AMS Eastern Sectional Meeting, Fall 2011 Sep. 12, 2011

2 Background Previous work Previous work relating self-similar fractals to boundaries [DS] Denker and Sato construct a Markov chain for which the Sierpinski gasket is (homeomorphic to) the Martin boundary. [Kai] Kaimanovich outlines a program for understanding the Sierpinski gasket as a Gromov boundary, by constructing an associated hyperbolic graph. [Kig2] Kigami develops the theory of resistance analysis on trees, obtains results on jump process on the boundary Cantor set. [LW] Lau and Wang extend Kaimanovich s construction to self-similar sets satisfying the open set condition. [JLW] Ju, Lau, and Wang extend Denker & Sato s work to a certain class of pcf self-similar fractals.

3 Background Networks and Markov chains Definition (Networks (G, c)) A network (G, c) is a connected simple graph. x G means x is a vertex. The edges are determined by a weight function called conductance: x y iff 0 < c xy <. x y iff c xy = 0. c is symmetric: c xy = c yx for all x, y G. Total conductance at a vertex must be finite: c(x) := c xy <, for all x G. y x

4 Background Networks and Markov chains Network energy and Laplacian Definition ((Dirichlet) energy of a function u : G R) E(u) := 1 2 c xy u(x) u(y) 2, dom E = {u. E(u) < }. x,y G c xy = 0 unless x y, so 1 2 means each edge is counted once. Note: Ker E = {constant functions}. For f : R n R, the continuous analogue is E(f ) := U f 2 dv.

5 Background Networks and Markov chains Network energy and Laplacian Definition ((Dirichlet) energy of a function u : G R) E(u) := 1 2 c xy u(x) u(y) 2, dom E = {u. E(u) < }. x,y G c xy = 0 unless x y, so 1 2 means each edge is counted once. Note: Ker E = {constant functions}. For f : R n R, the continuous analogue is E(f ) := U f 2 dv. Definition (Energy form E on functions u, v dom E) E(u, v) := 1 2 x,y G c xy(u(x) u(y))(v(x) v(y))

6 Background Networks and Markov chains Network energy and Laplacian Definition ((Dirichlet) energy of a function u : G R) E(u) := 1 2 c xy u(x) u(y) 2, dom E = {u. E(u) < }. x,y G c xy = 0 unless x y, so 1 2 means each edge is counted once. Definition (Laplace operator) The Laplacian is a symmetric operator acting on u : G R by ( u)(x) := c xy (u(x) u(y)) y x u is harmonic iff u = 0.

7 The network N F associated to a fractal F Construction of the network N F The IFS {Φ 0, Φ 1,..., Φ J } Φ j is a rigid motion of R d + homothety with scaling factor r j (0, 1). Ṽ 0 = {q 0, q 1,..., q J } are the fixed points of Φ j and Φ (A) := J Φ j (A) j=0 F is the unique nonempty compact subset with Φ(F) = F. Assume F carries a regular harmonic structure. (Need the self-similar energy form on F to have renomalization factors are < 1.)

8 The network N F associated to a fractal F Construction of the network N F The IFS {Φ 0, Φ 1,..., Φ J } Φ j is a rigid motion of R d + homothety with scaling factor r j (0, 1). Ṽ 0 = {q 0, q 1,..., q J } are the fixed points of Φ j and Φ (A) := J Φ j (A) j=0 F is the unique nonempty compact subset with Φ(F) = F. Assume F carries a regular harmonic structure. W m := {0, 1,..., J} m is the set of words w = w 1 w 2... w m. Φ w (x) := Φ w1 Φ w2... Φ wm (x) The set Φ w (F) is called a m-cell. Here w = m.

9 The network N F associated to a fractal F Construction of the network N F Discrete approximants of F Ṽ 0 = {q 0, q 1,..., q J } are the fixed points of Φ j. Γ 0 is the complete network on Ṽ 0. For now, c xy = 1. Define Γ k := Φ( Γ k 1 ) = Φ k ( Γ 0 ), so Γ k has vertices Ṽ k = Φ k (Ṽ 0 ) and edges c Φj (x)φ j (y) = c xy. Ṽ refers to vertices of F, V refers to vertices of N F. q 2 ~ ~ ~ G 0 G 1 G 2 F F q 0 q 1

10 The network N F associated to a fractal F Construction of the network N F The Sierpinski gasket SG and Sierpinski network N SG N SG F 2 G 2 F 1 G 1 F 0 G 0

11 The network N F associated to a fractal F Construction of the network N F Discrete approximants of F Ṽ 0 = {q 0, q 1,..., q J } are the fixed points of Φ j. Γ 0 is the complete network on Ṽ 0. For now, c xy = 1. Define Γ k := Φ( Γ k 1 ) = Φ k ( Γ 0 ), so Γ k has vertices Ṽ k = Φ k (Ṽ 0 ) and edges c Φj (x)φ j (y) = c xy. Ṽ refers to vertices of F, V refers to vertices of N F. The vertex set V of N F is a subset of F Z + : V k := Ṽ k {k} and Γ k := Γ k {k}. Then V := k=0 V k is the vertex set of N F.

12 The network N F associated to a fractal F Construction of the network N F The edges of N F Let E k denote the set of edges of the graph Γ k previously defined. The set of horizontal edges of N F is E = k=1 E k. q 2 E 0 E 1 E 2 q 0 q 1 F 1 F 2

13 The network N F associated to a fractal F Construction of the network N F The edges of N F Let E k denote the set of edges of the graph Γ k previously defined. The set of horizontal edges of N F is E = k=1 E k. The set of vertical edges of N F is F k : (Φ wj (q i ), k) (Φ w (q i ), k 1) w W k 1 and i, j {1, 2,..., J} Here, (Φ w (q i ), k 1) V k 1 and (Φ wj (q i ), k) V k. The set of all vertical edges is F = k=1 F k.

14 The network N F associated to a fractal F Construction of the network N F The Sierpinski gasket SG and Sierpinski network N SG N SG F 2 G 2 F 1 G 1 F 0 G 0

15 The network N F associated to a fractal F Compactifications and notions of boundary Compactifications If Ω C or R d, it is clear what Ω is. If Ω is the state space of a random walk, Ω describes limits. To obtain a boundary: use a family of functions U to obtain a compactification ˆΩ. Then Ω = ˆΩ \ Ω.

16 The network N F associated to a fractal F Compactifications and notions of boundary Compactifications If Ω C or R d, it is clear what Ω is. If Ω is the state space of a random walk, Ω describes limits. To obtain a boundary: use a family of functions U to obtain a compactification ˆΩ. Then Ω = ˆΩ \ Ω. Let U be a family of bounded functions. Then equivalently: (i) Take ˆΩ = Ω /, where Ω is the space of infinite paths in Ω and (x n) (y n) iff lim f (x n) = lim f (y n) for every f U. (ii) Take ˆΩ to be the smallest space to which each f U can be (continuously) extended. (iii) Take a metric completion, using d(x, y) = f U w f f (x) f (y), for weights w f > 0.

17 The network N F associated to a fractal F Compactifications and notions of boundary Compactifications If Ω C or R d, it is clear what Ω is. If Ω is the state space of a random walk, Ω describes limits. To obtain a boundary: use a family of functions U to obtain a compactification ˆΩ. Then Ω = ˆΩ \ Ω. For the Martin boundary, use U = {k(x, ). x Ω}, where k(x, y) = g(x, y) = E[#{X n = y} X 0 = x] = g(x, y) g(o, y). p n (x, y). n=0

18 The network N F associated to a fractal F Compactifications and notions of boundary Compactifications If Ω C or R d, it is clear what Ω is. If Ω is the state space of a random walk, Ω describes limits. To obtain a boundary: use a family of functions U to obtain a compactification ˆΩ. Then Ω = ˆΩ \ Ω. Theorem: For every harmonic and bounded function h on Ω, there is a Borel measure ν h on Ω such that h(x) = k(x, ) dν h, for every x Ω. Ω Next: we replace the Martin kernels with a family {u x } x G.

19 The network N F associated to a fractal F Bump functions at infinity Harmonically generated functions Define a function u on N F as follows: 1 Specify the values on V 0. 2 Let A w1,... A wj be the harmonic extension matrices. 3 Let Ψ j (x, z) = (Φ j (x), z + 1). 4 Define u on the remainder of N F via u ΨwV 0 = A w u V0. Lemma. The harmonically generated function u has finite energy. F E

20 The network N F associated to a fractal F Bump functions at infinity Harmonically generated functions Define a function u on N F as follows: 1 Specify the values on V 0. 2 Let A w1,... A wj be the harmonic extension matrices. 3 Let Ψ j (x, z) = (Φ j (x), z + 1). 4 Define u on the remainder of N F via u ΨwV0 = A w u V0. Lemma. The harmonically generated function u has finite energy. E(u) = E Ek (u) + E Fk (u) 2 r k E E0 (u) = E E 0 (u) 1 r <. k=0 k=1 k=0 Note: r < 1 follows from regularity assumption.

21 The network N F associated to a fractal F Bump functions at infinity Harmonically generated functions Define a function u on N F as follows: 1 Specify the values on V 0. 2 Let A w1,... A wj be the harmonic extension matrices. 3 Let Ψ j (x, z) = (Φ j (x), z + 1). 4 Define u on the remainder of N F via u ΨwV 0 = A w u V0. Lemma. The harmonically generated function u has finite energy. For x V 0, let u x denote the harmonically generated function initiated with { 1, y = x u x (y) = 0, y V 0 \ {x}.

22 The network N F associated to a fractal F Bump functions at infinity Localized harmonically generated functions For x = (ξ, m) N F, let K(ξ, m) be the union of the m-cells of F containing ξ. Repeat the construction of the harmonically generated function for u on the vertices below these cells, extend by 0 elsewhere: { u(ψ 1 (z)), z Ψ u x (z) := w (q) w (q)(n F ) for some q V 0, 0, else. lim u x(ξ, n) = 1, n limu x > 0 lim u x(ζ, n) = 0, for ζ / K(ξ, m). y = (ξ, m + 1). n limu x = 0 limu y > 0 K(x,m) limu y = 0 K(x,m+1) x x

23 The network N F associated to a fractal F Bump functions at infinity Localized harmonically generated functions For x = (ξ, m) N F, let K(ξ, m) be the union of the m-cells of F containing ξ. Repeat the construction of the harmonically generated function for u on the vertices below these cells, extend by 0 elsewhere: { u(ψ 1 (z)), z Ψ u x (z) := w (q) w (q)(n F ) for some q V 0, 0, else. Use the family U := {u x } x NF to compactify N F.

24 The boundary B F of the network N F Compactification of N F Path space in N F A path in N F is a sequence of adjacent vertices (x n ) n=1, x n = (ξ n, k n ) N F, ξ n F, k n Z +. The transition probabilities p(x, y) := c xy /c(x) induce a natural measures on the trajectory space Ω = {paths in N F }.

25 The boundary B F of the network N F Compactification of N F Path space in N F A path in N F is a sequence of adjacent vertices (x n ) n=1, x n = (ξ n, k n ) N F, ξ n F, k n Z +. The transition probabilities p(x, y) := c xy /c(x) induce a natural measures on the trajectory space Ω = {paths in N F }. Define an equivalence relation on Ω by (x n ) n=1 (y n ) n=1 lim u z(x n ) = lim u z(y n ), for all z G. n n where u z are harmonically generated. The boundary of N F is B F := Ω/, and N F = N F B F is a compactification of N F.

26 The boundary B F of the network N F Compactification of N F Path space in N F A path in N F is a sequence of adjacent vertices (x n ) n=1, x n = (ξ n, k n ) N F, ξ n F, k n Z +. The transition probabilities p(x, y) := c xy /c(x) induce a natural measures on the trajectory space Ω = {paths in N F }. P x is the corresponding measure on paths starting at x N F. Lemma. For any x N F, the random walk started at x is transient.

27 The boundary B F of the network N F Convergence to the boundary Theorem (Convergence to the boundary). There is a B F -valued r.v. X such that for every x N F, lim n X n = X, in the topology of N F. P x -almost surely,

28 The boundary B F of the network N F Convergence to the boundary Theorem (Convergence to the boundary). There is a B F -valued r.v. X such that for every x N F, lim n X n = X, P x -almost surely, in the topology of N F. In other words, if Ω := {(x n ) n=1 Ω. x n x B F in the topology of N F }, then: 1 Ω is measurable w.r. to A, the Borel σ-algebra of N F. 2 For every x N F, one has P x (Ω ) = 1. 3 For each x N F, the function X : Ω B F defined by X (ω) = x, for all ω = (x n ) Ω, is measurable with respect to A.

29 The boundary B F of the network N F Convergence to the boundary Theorem (Convergence to the boundary). There is a B F -valued r.v. X such that for every x N F, lim n X n = X, P x -almost surely, in the topology of N F. The main idea for the second part: For every x N F, one has P x (Ω ) = 1. For Martin boundary, convergence to the boundary comes from k(x, y) superharmonic = k(x n, y) supermartingale. For this construction, convergence to the boundary comes from u y (x) finite-energy = lim n u y (X n ) converges P x -a.e., by a key result in [ALP].

30 The boundary B F of the network N F Identification of the boundary Theorem (Identification of the boundary). The network boundary B F is (homeomorphic to) the attractor F. Idea of proof: Extend the u x functions to N F by continuity. These separate points of F, so can be used to establish a bijection f : F B F. For any point rational point ξ F, the family {K(ξ, m)} m m0 forms a neighbourhood basis at ξ. For ξ F and x V m, one has u x (f (ξ)) = ψ x (m) (ξ). Here ψ x (m) is the piecewise harmonic spline of [Str, Thm ]. For subbasic open sets U x,x,ε := {y B F... u x(x ) u x(y ) < ε}, show f 1 (U x,x,ε) is preimage of an open set under ψ (m) x. A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

31 The boundary B F of the network N F Harmonic extension Trace of the energy form Let Ω = Ω B and ϕ L (B). (Q): What is the energy of ϕ?

32 The boundary B F of the network N F Harmonic extension Trace of the energy form Let Ω = Ω B and ϕ L (B). (Q): What is the energy of ϕ? (A): Take the harmonic extension of ϕ to Ω: h ϕ (x) = E x (ϕ(x )) = ϕ dν x, for all x Ω. Now h ϕ is harmonic on Ω, so the energy-minimizing extension: h ϕ (x) = 0, for x Ω and h ϕ (x) = ϕ(x), for x B. B

33 The boundary B F of the network N F Harmonic extension Trace of the energy form Let Ω = Ω B and ϕ L (B). (Q): What is the energy of ϕ? (A): Take the harmonic extension of ϕ to Ω: h ϕ (x) = E x (ϕ(x )) = ϕ dν x, for all x Ω. Now h ϕ is harmonic on Ω, so the energy-minimizing extension: h ϕ (x) = 0, for x Ω and h ϕ (x) = ϕ(x), for x B. Define the trace energy form: B E B (ϕ) := E(h ϕ ), dom E B = {ϕ L (B). h ϕ dom E}.

34 The boundary B F of the network N F Harmonic extension Work in progress: Theorem (Harmonic extension). Let ϕ : F R satisfy E F (ϕ) <. Then for any x N F, there is a measure µ x on F such that h ϕ (x) = F ϕ(ξ) dµ x(ξ) defines a harmonic function of finite energy on N F. What conditions ensure a harmonic extension h ϕ? What conditions on h ensure a harmonic representation in terms of some ϕ? Can one establish a relation between E(h ϕ ) and E F (ϕ)? Here E F is the energy on F as defined in [Kig1] (see also [Str]). Recent related results: [WL, LN].

35 The boundary B F of the network N F Harmonic extension Alano Ancona, Russell Lyons, and Yuval Peres. Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. Ann. Probab. 27(1999), Manfred Denker and Hiroshi Sato. Sierpiński gasket as a Martin boundary. I. Martin kernels. Potential Anal. 14(2001), Hongbing Ju, Ka-Sing Lau, and Xiang-Yang Wang. Post-critically finite fractals and Martin boundary. To appear: Trans. AMS (2009), Vadim A. Kaimanovich. Random walks on Sierpinski graphs: hyperbolicity and stochastic homogenization. In Fractals in Graz 2001, Trends Math., pages Birkhäuser, Basel, 2003.

36 The boundary B F of the network N F Harmonic extension Jun Kigami. Analysis on fractals, volume 143 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, Jun Kigami. Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees. Adv. Math. 225(2010), Ka-Sing Lau and Sze-Man Ngai. Martin boundary and exit space on the Sierpinski gasket and other fractals. Preprint (2011), Ka-Sing Lau and Xiang-Yang Wang. Self-similar sets as hyperbolic boundaries. Indiana Univ. Math. J. 58(2009),

37 The boundary B F of the network N F Harmonic extension Robert S. Strichartz. Differential equations on fractals. Princeton University Press, Princeton, NJ, A tutorial. Ting-Kam Wong and Ka-Sing Lau. Random walks and induced Dirichlet forms on self-similar sets. Preprint (2011), 1 26.