Quasi-Harmonic Functions on Finite Type Fractals

Transcription

1 Quasi-Harmonic Functions on Finite Type Fractals Nguyen Viet Hung Mathias Mesing Department of Mathematics and Computer Science University of Greifswald, Germany Workshop Fractal Analysis

2 Outline 1 Introduction 2 Fractals of finite type 3 Constructing recursive functions 4 Example Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

3 Notations Let f i : R d R d contractions with same ratio r, i S = {1,..., m}, and E the belonging invariant set. Set S n := {(u i ) i=1,...,n u i S i = 1,..., n}, S := n N S n. For u := u 1... u n S define f u := f u1... f un and E u := f u (E). Example (Christmas tree fractal) f i : C C, i = 1, 2, 3 f i (z) = a(z + c i ) with c i = e 2π(i 1) 3, a = 1/( i 3 2 ) Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

4 Motivation Harmonic functions on Sierpinski gasket New approach Regard functions on points of E. Calculate the values at new points by averaging rules: g(x) = g(a)+2g(b)+2g(c) 5 g(y) = 2g(a)+g(b)+2g(c) 5 g(z) = 2g(a)+2g(b)+g(c) 5 Regard functions on subpieces of E. Use averaging rules to calculate the values on smaller pieces. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

5 Motivation Harmonic functions on Sierpinski gasket New approach Regard functions on points of E. Calculate the values at new points by averaging rules: g(x) = g(a)+2g(b)+2g(c) 5 g(y) = 2g(a)+g(b)+2g(c) 5 g(z) = 2g(a)+2g(b)+g(c) 5 Regard functions on subpieces of E. Use averaging rules to calculate the values on smaller pieces. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

6 Neighbors and neighborhood Example (Sierpinski Gasket) Neighbors of a subpiece E u All subpieces E v with u = v E u E v E u E v Neighborhood of a subpiece E u The set of all neighbors of E u. Neighborhood type Neighborhood up to similarity. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

7 Neighbors and neighborhood Example (Sierpinski Gasket) Neighbors of a subpiece E u All subpieces E v with u = v E u E v E u E v Neighborhood of a subpiece E u The set of all neighbors of E u. Neighborhood type Neighborhood up to similarity. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

8 Neighbors and neighborhood Example (Sierpinski Gasket) Neighbors of a subpiece E u All subpieces E v with u = v E u E v E u E v Neighborhood of a subpiece E u The set of all neighbors of E u. Neighborhood type Neighborhood up to similarity. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

9 Finite type Definition A fractal of finite type is one with only finitely many neighborhood types. Example (Christmas tree fractal) The Christmas tree fractal is of finite type. Example (Kenyon) The following example given by Kenyon is not of finite type. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

10 Finite type Definition A fractal of finite type is one with only finitely many neighborhood types. Example (Christmas tree fractal) The Christmas tree fractal is of finite type. Example (Kenyon) The following example given by Kenyon is not of finite type. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

11 Finite type Definition A fractal of finite type is one with only finitely many neighborhood types. Example (Christmas tree fractal) The Christmas tree fractal is of finite type. Example (Kenyon) The following example given by Kenyon is not of finite type. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

12 Post-critically finite vs. finite type Example post-critically finite finite type Boundary contains finitely many could contain infinitely points many points E i E j contains finitely many could contain infinitely points many points parallel paths few many Analysis well-developed only first steps Open question: post-critically finite finite type? Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

13 Neighbor maps Definition (Bandt) Let E u and E v be neighbors. Then the map h uv := fu 1 f v is called neighbor map. In this case we say E u and E v are of neighbor type T (u, v). Example (Sierpinski Gasket) Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

14 Neighbour graph I Definition The neighbor graph is a pair N = (V, E) with V := {h uv h uv is a neighbor map} {id} E := {(h uv, h xy ) V V i, j S with T (x, y) = T (ui, vj)} Example (Sierpinski Gasket) h12 h h id h31 13 h 23 h Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

15 Neighborgraph II Properties The neighborgraph contains information about the neighbor pieces and their relative position the neighborhood types of the fractal the topological structure of the fractal Example (Christmas tree fractal) Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

16 Neighborgraph III Theorem (Bandt, Mesing) The sets E i E j contain finitely many points iff there are only disjoint circles in N and there are no edges joining one circle with another. The sets E i E j contain uncountable many points iff at least two circles in G have at least a common vertex. Example (Sierpinski gasket) Example (Christmas tree fractal) Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

17 Neighborgraph III Theorem (Bandt, Mesing) The sets E i E j contain finitely many points iff there are only disjoint circles in N and there are no edges joining one circle with another. The sets E i E j contain uncountable many points iff at least two circles in G have at least a common vertex. Example (Sierpinski gasket) h12 h 32 Example (Christmas tree fractal) 31 h id h31 13 h 23 h Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

18 Neighborgraph III Theorem (Bandt, Mesing) The sets E i E j contain finitely many points iff there are only disjoint circles in N and there are no edges joining one circle with another. The sets E i E j contain uncountable many points iff at least two circles in G have at least a common vertex. Example (Sierpinski gasket) h12 h 32 Example (Christmas tree fractal) 31 h id h31 13 h 23 h Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

19 Analysis on Sierpinski gasket Building harmonic function on Sierpinski gasket by points. f (x) = f (y) = f (z) = 2f (a)+2f (b)+f (c) 5 2f (a)+f (b)+2f (c) 5 f (a)+2f (b)+2f (c) 5 Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

20 Step 1, Starting values G(E) = G(A) + G(B) + G(C) 3 Step 2, calculating the values on sub pieces Formula, λ = 1/4 G(X 1) = G(X ) + λg(y ) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

21 Step 1, Starting values G(E) = G(A) + G(B) + G(C) 3 Step 2, calculating the values on sub pieces Formula, λ = 1/4 G(X 1) = G(X ) + λg(y ) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

22 Step 1, Starting values G(E) = G(A) + G(B) + G(C) 3 Step 2, calculating the values on sub pieces Formula, λ = 1/4 G(X 1) = G(X ) + λg(y ) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

23 Preparation From now on, we regard the connected finite type fractal E as a part of the network, with type T, i.e E has k T -neighbor h 1 (E)...h kt (E). Example Some definition For any infinite word u, denote u n := u 1...u n. S 0 := { } Define the extension of the set S n as S n := S n N n S 1 for Christmas tree fractal. where, N n := {(v, k) h k (E v ) E, v = n, k = 1..k T } S := n N S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

24 Preparation From now on, we regard the connected finite type fractal E as a part of the network, with type T, i.e E has k T -neighbor h 1 (E)...h kt (E). Example Some definition For any infinite word u, denote u n := u 1...u n. S 0 := { } Define the extension of the set S n as S n := S n N n S 1 for Christmas tree fractal. where, N n := {(v, k) h k (E v ) E, v = n, k = 1..k T } S := n N S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

25 Preparation From now on, we regard the connected finite type fractal E as a part of the network, with type T, i.e E has k T -neighbor h 1 (E)...h kt (E). Example Some definition For any infinite word u, denote u n := u 1...u n. S 0 := { } Define the extension of the set S n as S n := S n N n S 1 for Christmas tree fractal. where, N n := {(v, k) h k (E v ) E, v = n, k = 1..k T } S := n N S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

26 Preparation From now on, we regard the connected finite type fractal E as a part of the network, with type T, i.e E has k T -neighbor h 1 (E)...h kt (E). Example Some definition For any infinite word u, denote u n := u 1...u n. S 0 := { } Define the extension of the set S n as S n := S n N n S 1 for Christmas tree fractal. where, N n := {(v, k) h k (E v ) E, v = n, k = 1..k T } S := n N S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

27 Goal Main goal We would like to construct and study a class of functions which behave similar to Harmonic functions on E Idea This class is built based on a class of functions G : S R satisfying three conditions: lim n G(u n ) exist for all infinite words u. lim n G(u n ) = lim n G(v n ) for any u, v S : π(u) = π(v), where π the projection from S to E. The corresponding map g determined by g(x) := {lim n G(u n ) π(u) = x} on E has some nice properties. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

28 Goal Main goal We would like to construct and study a class of functions which behave similar to Harmonic functions on E Idea This class is built based on a class of functions G : S R satisfying three conditions: lim n G(u n ) exist for all infinite words u. lim n G(u n ) = lim n G(v n ) for any u, v S : π(u) = π(v), where π the projection from S to E. The corresponding map g determined by g(x) := {lim n G(u n ) π(u) = x} on E has some nice properties. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

29 Method Step 1, Setting parameter. For any n N, u, v S n, each letter w S, we assign one parameters λ(w, T (u, v)) [0, 1). Denote Λ be the finite set containing all λ(w, T (u, v)). Step 2, Structure of recursive functions Assume that the values of G on S 0 are given. We construct the recursive functions G by recurrence: G(uw) := G(u) + v N (u) λ(w, T (u, v))g(v) 1 + v N (u) λ(w, T (u, v)) for any n N, w S, u, v S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

30 Method Step 1, Setting parameter. For any n N, u, v S n, each letter w S, we assign one parameters λ(w, T (u, v)) [0, 1). Denote Λ be the finite set containing all λ(w, T (u, v)). Step 2, Structure of recursive functions Assume that the values of G on S 0 are given. We construct the recursive functions G by recurrence: G(uw) := G(u) + v N (u) λ(w, T (u, v))g(v) 1 + v N (u) λ(w, T (u, v)) for any n N, w S, u, v S n. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

31 Recursive formula,(calculating values on S n+1 base on S n ) G(uw) := G(u) + v N (u) λ(w, T (u, v))g(v) 1 + v N (u) λ(w, T (u, v)) Example (Sierpinski gasket) G(X 1) = G(X ) + λ G(Y ) + 0 G(Z) + 0 G(Q) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

32 Λ assumption For any letter w S, any n N, any u, v S n, λ(w, T (u, v))p(w) < min λ(w, T (u, v)) v uw v uw where P(w) := 1 + v N (u),w S λ(w, T (u, v)). Example (Sierpinski gasket) G(X 1) = G(X ) + λ G(Y ) + 0 G(Z) + 0 G(Q) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

33 General results Theorem (Existence) Suppose that Λ satisfies the Λ assumption.then for any starting values of recursive function G on S 0, the multi-valued map g is a well-defined function on E. Let H(Λ) be the class of all functions g in the existence theorem. Theorem (Continuity ) H(Λ) is a subspace of the continuous function space on E. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

34 General results Theorem (Existence) Suppose that Λ satisfies the Λ assumption.then for any starting values of recursive function G on S 0, the multi-valued map g is a well-defined function on E. Let H(Λ) be the class of all functions g in the existence theorem. Theorem (Continuity ) H(Λ) is a subspace of the continuous function space on E. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

35 Quasi-harmonic condition For all recursive functions G, G( ) = α i G((, i)), where : empty word. i=1..k T for some (α i ) k T i=1 (0, 1) satisfying α α kt = 1 Theorem (Maximum Principle) Assume that the quasi-harmonic condition is hold. Then all functions g H(Λ) satisfy Maximum principle. i.e: max{g(x) x E} max{g(x) x B(E)} (min{g(x) x E} min{g(x) x B(E)}) and if the maximum (or minimum) obtains on E \ B(E) then g is constant. We call functions belonging to H(Λ), for some Λ, quasi-harmonic. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

36 Sierpinski gasket Λ = {λ} There is only one type by symmetry: G(X 1) = G(X ) + λg(y ) 1 + λ Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

37 Theorem (Hölder continuity) For λ (0, 1), H(Λ) contains α Hölder continuous functions, where α = log(λ + 1) log(max(λ, 1 λ)). log(2) Then, α (1, ln 3 ln 2 ]. Corollary (Differentiability) All the functions in H(Λ) are differentiable. Proposition Assume that G( ) = 1 3 [G((, 1)) + G((, 2)) + G((, 3))] for all recursive functions G. Then, for λ = 1/4, H(Λ) is the class of harmonic functions on E. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

38 Theorem (Hölder continuity) For λ (0, 1), H(Λ) contains α Hölder continuous functions, where α = log(λ + 1) log(max(λ, 1 λ)). log(2) Then, α (1, ln 3 ln 2 ]. Corollary (Differentiability) All the functions in H(Λ) are differentiable. Proposition Assume that G( ) = 1 3 [G((, 1)) + G((, 2)) + G((, 3))] for all recursive functions G. Then, for λ = 1/4, H(Λ) is the class of harmonic functions on E. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

39 Theorem (Hölder continuity) For λ (0, 1), H(Λ) contains α Hölder continuous functions, where α = log(λ + 1) log(max(λ, 1 λ)). log(2) Then, α (1, ln 3 ln 2 ]. Corollary (Differentiability) All the functions in H(Λ) are differentiable. Proposition Assume that G( ) = 1 3 [G((, 1)) + G((, 2)) + G((, 3))] for all recursive functions G. Then, for λ = 1/4, H(Λ) is the class of harmonic functions on E. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23

40 Summary Purpose: Using the neighbor graph to construct quasi-harmonic functions. Advantage: These functions are not only determined on p.c.f fractals but also on finite type fractals. Disadvantage: The space of functions is a finite dimensional space. Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis / 23