Self-similar Sets as Quotients of Shift Spaces Jun Kigami Kyoto University

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1 Self-similar Sets as Quotients of Shift Spaces Jun Kigami Kyoto University

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3 (One-sided) Shift space S: a finite set, symbols For w = w 1...w m W, Σ = S N = {w 1 w 2... i N,w i S} W m = S m = {w 1...w m w 1,...,w m S} W = W m m 0 σ(ω 1 ω 2 ω 3...)= ω 2 ω 3...: Shift map Σ w = {ι ι = ι 1 ι 2... Σ, ι 1...ι m = w 1...w m }

4 Self-Similar sets (X, d ): a complete metric space {F s } s S : a collection of contractions on X, where F : X X is a contraction. LipF = sup x y X d (F (x),f(y)) d (x, y) < 1. Notations For ω = ω 1 ω 2... Σ and A X, F ω1...ω m A ω1...ω m = F ω1... F ωm = F ω1...ω m (A).

5 Theorem K: non-empty compact which satisfies K = s S F s (K) and a continuous surjection π : Σ K such that ω = ω 1 ω 2... Σ, s S, {π(ω)} = m 0 K ω1...ω m π(sω) =F s (π(ω)). K: the self-similar set associated with {F s } s S

6 Self-similar set = quotient of shift space K = Σ/π Fact: Let (w) = wwww... Σ for w W. Then F w (π((w) )=π((w) ) Assumption sup x K #(π 1 (x)) < + π((w) ) = the fixed point of F w.

7 C = K s1 K s2 : Critical set s 1 s 2 S P = m 1 σ m (C) : Post-critical set V 0 = π(p) Proposition 0.2. If w, v W with Σ w Σ v =, then K w K v = F w (V 0 ) F v (V 0 ).

8 Example: the Sierpinski gasket

9 Example: the unite square

10 The Sierpinski cross: r = 2 5

11 Intersection type IP = {(w, v) w, v W, Σ w Σ v =,K w K v } : Intersecting Pairs For (w, v) IP, ine ψ (w,v) :(F w ) 1 (K w K v ) (F v ) 1 (K w K v )by ψ (w,v) =(F v ) 1 F w (Fw ) 1 (K w K v ) ψ (w,v) : the intersection type of a intersecting pair (w, v). IT = {ψ (w,v) (w, v) IP}

12 Problems: (1) Given (r s ) s S (0, 1) S,? a metric d under which diam(k w1...w m,d) r w1...w m, where r w1...w m = r w1 r wm and diam(a, d) = sup x,y A d(x, y). (2) When a measure µ has the volume doubling property with respect to a metric d? Definition µ has the volume doubling property with respect to d C >0 such that x K, r >0, where B d (x, r) ={y d(x, y) <r}. µ(b d (x, 2r)) Cµ(B d (x, r)), (3) Construction of a diffusion process

13 Gauge functions = sizes of K w s balls U g (x, s) and metrics D g (x, y) Definition g : W [0, 1] is a gauge function. (1) g(φ) =1, g(wi) g(w) if w W and i S. (2) lim m Examples (1) r =(r s ) s S (0, 1) S, sup g(w) =0. w W m g r (w 1...w m )= r w1...w m : self-similar gauge function

14 (2) D(K) = {d d is a metric on K, diam(k, d) =1 (K, d) is homeomorphic to (K, d ).} For d D(K), ine d : W [0, 1] by d(w) = diam(k w,d). (3) M(K) = {µ µ: Borel regualr probability measure on K, w W,µ(K w ) > 0, finite set A, µ(a) =0.} For µ M(K), ine µ : W [0, 1] by µ(w) =µ(k w ). Abuse of notations!!

15 Definition 0.3. Λ g s = {w 1...w m W g(w 1...w m 1 ) >s g(w 1...w m )} = the collection of w s with g(w) s, Λ g s(x) = {w w Λ g s(x), v Λ g s such that x K v and K w K v } U g (x, s) = K w w Λ g s(x) = ball around x with radius s

16 Good metric = Adapted metric d D(K) is adapted to g c 1,c 2 > 0, x K, r >0 U g (x, c 1 r) B d (x, r) U g (x, c 2 r) Proposition 0.4. d D(K): adapted to g d is adapted to d d D(K) is adapted d is adapted to d.

17 Definition 0.5. A gauge function g is elliptic λ (0, 1), m, w W, v W m, i S, if wv, wi T, then g(wv) λg(w) g(wi) A gauge function g is locally finite sup x K,s (0,1] #(Λ g s(x)) < +. A gauge function g is called intersection type finite #(IT (g)) < +, where IT (g) ={ψ (w,v) (w, v) IP, s (0, 1],w,v Λ g s}.

18 Construction of a metric from a gauge function For a gauge function g, ine { m } D g (x, y) = inf g(w(i)) m 1, (w(1),...,w(m)): a path x y i=1 D g (x, y): symmetric, non-negative, satisfy the triangle inequality but x y D g (x, y) > 0?? Theorem 0.6. g is elliptic and intersection type finite α (0, 1], D g α(x, y): a metric on K adapted to g α Theorem 0.7. If K is rationally ramified and g is elliptic, then intersection type finite locally finite

19 Definition 0.8. g, h: gauge functions, h GE g c >0 w, v Λ s g if K w K v, then h(w) ch(v) Proposition 0.9. (1) GE is an equivalence relation on elliptic gauge functions. (2) Among elliptic gauge functions, being locally finite and intersection type finite are invariant under GE.

20 Theorem Assume d is a good metric, i.e. d is adapted to d and d is locally finite. Then µ M(K) has the volume doubling property with respect to d µ is elliptic and d GE µ.

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