From self-similar groups to intrinsic metrics on fractals

Transcription

1 From self-similar groups to intrinsic metrics on fractals *D. J. Kelleher 1 1 Department of Mathematics University of Connecticut Cornell Analysis Seminar Fall 2013

2 Post-Critically finite fractals In the case where the fractal is post-critically finite, a good deal is known about the analysis on these structures 1 The existence of a Dirichlet form 2 Equivalently, the existence of a Laplacian operator or a symmetric Hunt process.

3 Post-Critically finite fractals In the case where the fractal is post-critically finite, a good deal is known about the analysis on these structures 1 Jun Kigami - Analysis on Fractals Robert Strichartz - Differential Equations on Fractals Jun Kigami - Memoirs of the AMS (Resistance forms)

4 Part I: From self-similar groups From Self-Similar Structures to Self-Similar Groups IJAC Vol. 22, No. 07 (2012) with Ben Steinhurst and Mike Wong. Idea Determine the relationship between dynamical systems arising from self-similar groups and self-similar fractals.

5 Rooted Trees and Self-Similar Actions (Nekrashevych) A finite set X, called the alphabet Rooted tree structure of X, the set of all words An automorphism of X preserves adjacency of vertices A self-similar group G is a subgroup of Aut X that acts on X letter by letter G contracting if can be represented by a finite Moore diagram

6 Moore Diagrams and Limit Spaces (Nekrashevych) Example: Binary adding machine X = {0, 1}, a(0) = 1, a(1) = 0, G = a (0, 0) (0, 1) (1, 0) (0, 1) a 1 e a (1, 0) (1, 1) Figure : Binary adding machine a(11001) = 0 a(1001) = 00 a(001) = 001 e(01) = 00101

7 Moore Diagrams and Limit Spaces (Nekrashevych) Example: Binary adding machine X = {0, 1}, a(0) = 1, a(1) = 0, G = a (0, 0) (0, 1) (1, 0) (0, 1) a 1 e a (1, 0) (1, 1) Figure : Binary adding machine Left-infinite paths define an asymptotic equivalence relation G on X ω ; J G = X ω / G is the limit space of G Binary adding machine: (read from the right) 01w G 10w Limit space is the circle

8 Moore Diagrams and Limit Spaces (Nekrashevych) Example: Binary adding machine X = {0, 1}, a(0) = 1, a(1) = 0, G = a (0, 0) (0, 1) (1, 0) (0, 1) a 1 e a (1, 0) (1, 1) Figure : Binary adding machine 1 The asymptotic equivalence relation is invariant under the right-shift, s, that is x 3 x 2 x 1 G y 3 y 2 y 1 x 3 x 2 G y 3 y 2 2 In particular, s descends to a self-cover of J G.

9 Hanoi Towers Group and Sierpiński Gasket Hanoi Towers Group: (2, 2) a 01 (0, 1) (1, 0) (2, 1) 1 (0, 2) a 12 (1, 2) (2, 0) a 02 (0, 0) (1, 1) Figure : Hanoi Towers Group

10 Hanoi Towers Group and Sierpiński Gasket Hanoi Towers Group: (2, 2) Limit space: Sierpiński Gasket a 01 (0, 1) (1, 0) (2, 1) 1 (0, 2) a 12 (1, 2) (2, 0) a 02 (0, 0) (1, 1) Figure : Hanoi Towers Group Figure : Sierpiński Gakset

11 Self-Similar Structures (Kigami) F i : K K continuous injection for each i X, mapping K to a smaller part of itself A surjection π : X ω K from the code space X ω to K, marking the image of F i by i L = (K, X, {F i } i X ) is a self-similar structure on K For a point a K, π 1 (a) contains the addresses of a Example: Sierpiński Gasket (Usual Structure) Figure : Sierpiński Gasket

12 Motivation Question: Given a P.c.f. self-similar structure, when can we find a contracting group that produces a limit space with this structure, and visa versa? Necessary condition: If π(... x 2 x 1 ) = π(... y 2 y 1 ), then π(... x n+1 x n ) = π(... y n+1 y n ) for all n Equivalently: the induced shift map s : J G J G defined by s = Fi 1 for each image of F i, exists and is continuous Necessary condition: If π(... x 2 x 1 ) = π(... y 2 y 1 ), then π(... x 2 x 1 w) = π(... y 2 y 1 w) for all w X n Because applying F i (π(... x 2 x 1 )) = π(... x 2 x 1 i) for any i X.

13 When Does a Limit Space Have a Self-Similar Structure? A self-similar structure L = (J G, X, {F i } i X ) on a limit space J G, such that p = π Limit space of the binary adding machine, the circle, does not have a self-similar structure Theorem (1) The limit space J G has a self-similar structure if and only if it satisfies the following condition: For every left-infinite path e =... e 2 e 1 in the nucleus ending at a non-trivial state and for every w X, there exists a left-infinite path f =... f 2 f 1 in the nucleus ending at a state g, such that the label of the edge f n is the same as the label of e n, and g(w) = w.

14 When Does a Limit Space Have a P.C.F. Self-Similar Structure? (Slide I) Limit space: finitely ramified in the group-theoretical sense if the intersection of distinct tiles of the same level is finite Self-similar structure: finitely ramified in the fractal sense if the intersection of the images of F i is finite post-critically finite (p.c.f.) if 1 the set of addresses of the intersection of F i is finite 2 each address in this set has a recurring tail Significance: Can define Laplacian on the space (Bondarenko and Nekrashevych) A contracting group G is p.c.f. if there exists a finite number of left-infinite paths in its nucleus that end at a non-trivial state

15 When Does a Limit Space have a P.C.F. Self-Similar Structure? (Slide II) Lemma (Bondarenko and Nekrashevych 2003) The limit space J G is finitely ramified in the group-theoretical sense if and only if G is p.c.f. Theorem (2) The self-similar structure L = (J G, X, {F i } i X ) on the limit space J G of a contracting G is p.c.f. if and only if G is p.c.f. Point 1: Finitely ramified in the group-theoretical sense is the same as in the fractal sense when J G has a self-similar structure Point 2: Justifies use of the term p.c.f. group

16 When Does a Limit Space have a P.C.F. Self-Similar Structure? (Slide III) Corollary A limit space J G has a p.c.f. self-similar structure if and only if G satisfies the condition in Theorem (1) and is p.c.f. Corollary The self-similar structure on a limit space is p.c.f. if and only if it is finitely ramified.

17 From Kigami to Nekrashevych Theorem A p.c.f. self similar structure L = (K, X, {F i } i X ) is the limit space of a space of a p.c.f. self-similar if and only if there is a self covering s : K K such that s π = π σ. A self-similar group can be constructed throught the iterated monodromy group. We give a proof with a more elementary construction based on the equivalences classes induced by π.

18 Construction (Slide I) For a p.c.f. self-similar structure L satisfying the necessary condition, π(... x 2 x 1 ) = π(... y 2 y 1 ) implies that 1 π(... x n+1 x n ) = π(... y n+1 y n ) for all n Z + 2 π(... x 2 x 1 w) = π(... y 2 y 1 ) for all w X Write down the equivalence classes induced by L systematically: By (1) and (2), π(... x N+1 x N w) = π(... y N+1 y N w) where x N y N, which accounts for the original equation. Therefore, assume x 1 y 1. Then... x 2 x 1,... y 2 y 1 C. C is finite, so there are only finitely many such equations. L p.c.f. implies that elements in C have a recurring tail, so we can write π(zx n... x 2 x 1 w) = π(zy n... y 2 y 1 w). Also, z k x n or y n, and z is the shortest recurring word.

19 Construction (Slide II) Write down the equivalence classes induced by L systematically (continued): By (2) and induction, if π(zx n... x 2 x 1 w) = π(zy n... y 2 y 1 w), then π(zξ n... ξ 2 ξ 1 w) = π(zx n... x 2 x 1 w) whenever ξ j {x j, y j } for all j. Therefore, we can write all equivalence classes in the form {zζ n... ζ 2 ζ 1 w z, w X, ζ j S j } for fixed W, z X and some S j X; we introduce a shorthand to denote this: zs n... S 2 S 1 w. Notice that each equivalence class is determined by zs n... S 2 S 1 = π 1 (α) for some α π(c), where S 1 > 1. We use α to label zs n... S 2 S 1.

20 Construction (Slide III) For each zs n... S 2 S 1 = π 1 (α), we define some generators: (z 2, z 2) (z k 1, z k 1 ) g n+1 g n+k 1 (z 1, z 1) g n g n 1 g 1 1 (i, σ Sn (i)) (i, σ S2 (i)) (i, σ S1 (i)) (z k, z k ) Figure : The generators corresponding to α = Ψ(zS n... S 2 S 1 w) The desired group G L is the group generated by all the generators defined above for all α π(c) Theorem G L produces a self-similar structure L that is isomorphic to L.

21 Example: Unit Interval For the usual self-similar structure, the induced shift map s does not exist! A twisted structure: I = [0, 1] F 0 (x) = (1/2)x + 1/2, F 1 (x) = (1/2)x + 1/2 All equivalence classes determined by the equivalence class π 1 (1/2) = 1S 2 S 1, where S 2 = {0} and S 1 = {0, 1}, i.e. 100w 101w. We define the group as follows: (0, 1) (1, 1) g 2 g 1 1 (0, 0) (1, 0) Figure : The generators of G L Compare with the Grigorchuk group

22 Example: Sierpiński Gasket Figure : Sierpiński Gasket (Twisted Structure) s does not exist for usual structure; need a twisted structure Construction by our method: Yields Hanoi Towers Group!

23 Example: Pentakun Figure : Pentakun s does exist, so our construction yields a group

24 Example: Pentakun Figure : Pentakun s does exist, so our construction yields a group

25 Example: Snowflake Figure : Snowflake s does not exist, so not possible to find a group

26 Non-p.c.f. structures Less is known about analysis on fractals which aren t p.c.f. 1 The main results are for the Sierpinski carpet 2 There is a unique harmonic structure, but it is harder to get at.

27 Part II: Barycentric subdivisions and the hexacarpet Random walks on barycentric subdivisions and Strichartz hexacarpet, Experimental Mathematics. 21(4) 2012, Joint work with Matt Begue, Aaron Nelson, Hugo Panzo, Alexander Teplyaev. Also Antoni Brzoska, Diwakar Raisingh, and Gabe Khan

28 Barycentric subdivision We start off with a simplicial complex, Take each 1-simplex (line segment) and add a vertex at the midpoint. Take each 2-simplex (triangle) add a vertex at the barycenter (average of corners), ad an edge connecting the barycenter to each vertex adjacent to the original 2-simplex.... proceed inductively.

29 Start off with a simplex

30 Add midpoints to simplexes

31 Add edges to simpleces

32 We approximate the hexacarpet with a graph that has as vertices the faces of the nth barycentric subdivision, which are connect if the faces share and edge. (the second level is pictured above)

33 This is what the 4th level approximation looks like.

34 To get a fractal out of this, we looks at the symbolic dynamic system Let X = {0, 1,..., 5} where x is any element in X and v {0, 5} ω. Suppose i is odd and j = i + 1 mod 6. Then, xi3v xj3v and xi4v xj4v. (1) If i is even (j is still i + 1 mod 6), then xi1v xj1v and xi2v xj2v. (2) This produces a fractal with Cantor boundaries and hexagonal symmetries. We call it the hexacarpet.

35 (a) (ϕ2, ϕ3) (b) (ϕ2, ϕ4) (c) (ϕ2, ϕ5) (d) (ϕ2, ϕ6) (e) (ϕ3, ϕ4) (f) (ϕ3, ϕ5) (g) (ϕ3, ϕ6) (h) (ϕ4, ϕ5) (i) (ϕ4, ϕ6)

36 (a) (ϕ 2, ϕ 3, ϕ 4) (b) (ϕ 2, ϕ 3, ϕ 5) (c) (ϕ 2, ϕ 3, ϕ 6) (d) (ϕ 2, ϕ 3, ϕ 7) (e) (ϕ 2, ϕ 4, ϕ 5) (f) (ϕ 2, ϕ 4, ϕ 6) (g) (ϕ 2, ϕ 5, ϕ 6) (h) (ϕ 3, ϕ 5, ϕ 6) (i) (ϕ 4, ϕ 5, ϕ 6)

37 Level n c Table : Hexacarpet estimates for resistance coefficient c given by 1 λ n j. 6 λ n+1 j

38 Eigenvalue counting function The function N(x) = # {λ < x λ σ( )} for the 7th level approximating graph.

39 Higher dimensions

40 Higher dimensions We can perform a construction analogous to the hexacarpet on the 3-simplex. This time, our approximating graphs have tetrahedra as verteces, connected if they share a face Figure : First and second level graph approximations to the 3-simplectic sponge.

41 Eigenfunction Pictures - Level 4

42 Numerics Original Shape Triangle Tetrahedron Subdivisions (N) 6 24 Hausdorff dimension log(6) log(2) 2.58 log(24) log(2) Resistance scaling (ρ) Spectral dimension

43 Conjectures For each of the barycentric fractals 1 there exists a unique self-similar local regular conservative Dirichlet form E with resistance scaling factor ρ and the Laplacian scaling factor τ = 6ρ. 2 the simple random walks on the repeated barycentric subdivisions of a triangle, with the time renormalized by τ n, converge to the diffusion process, which is the continuous symmetric strong Markov process corresponding to the Dirichlet form E. 3 the spectral zeta function has a meromorphic continuation to C.

44 Conjectures For each of the barycentric fractals 1 there exists a unique self-similar local regular conservative Dirichlet form E with resistance scaling factor ρ and the Laplacian scaling factor τ = 6ρ. 2 the simple random walks on the repeated barycentric subdivisions of a triangle, with the time renormalized by τ n, converge to the diffusion process, which is the continuous symmetric strong Markov process corresponding to the Dirichlet form E. 3 the spectral zeta function has a meromorphic continuation to C.

45 Conjectures For each of the barycentric fractals 1 there exists a unique self-similar local regular conservative Dirichlet form E with resistance scaling factor ρ and the Laplacian scaling factor τ = 6ρ. 2 the simple random walks on the repeated barycentric subdivisions of a triangle, with the time renormalized by τ n, converge to the diffusion process, which is the continuous symmetric strong Markov process corresponding to the Dirichlet form E. 3 the spectral zeta function has a meromorphic continuation to C.

46 Heat kernel estimates Barlow and Bass used a probabilaistical interperation, comparing transition density of a random walk to the diffusion of heat, to prove, in some cases, that the heat kernel can be approximated for time 0 < t 1 p(t, x, y) t ds/2 exp ( c R(x, y) dw dw 1 t 1 dw 1 Where R is the effective resistance metric, and d h = Hausdorff dimension d s = Spectral dimension d w = 2d h d s = Walk dimension 2 )

47 Heat kernel p(t, x, y) t ds/2 exp ( c R(x, y) dw dw 1 t 1 dw 1 Where R is the effective resistance metric, and d h = Hausdorff dimension d s = Spectral dimension d w = 2d h d s = Walk dimension 2 ) However, the Hexacarpet and higher dimensional analogues do not quite fit this frame. There may be some logarithmic corrections...

48 If we look at the graph approximation, the inner circle of the graph has length 3 2 n at the nth level. The perimeter of this graph will have 3n2 n length. How do we resize the graph?

49 Another fractal What if we consider the triangle, and then perform barycentric subdivision. This time, we keep the simplicial complex, but renormalize each triangle to be equilateral. In the case of the first subdivision, we get a hexagon.

50 If we renormalize so that each line segment has length 2 n on the nth level, the induced geodesic metric converges to a metric on the triangle. 1 The Hausdorff (and self-similarity) dimension of this object is log 2 (6) Topologically we still have the same object! i.e. the limit is homeomorphic to the Euclidean triangle. 3 The distance from a vertex of an nth level simplex to the any point on the opposite side is 2 n. 4 There are infinitely many geodesics between points. The same construction can be done when starting with any n-simplex.

51 Part III: Intrinsic metrics on fractals Metrics and Spectral Triples for Dirichlet and Resistance Spaces To appear in Journal of Noncommutative Geometry, and Measures and Dirichlet forms under the Gelfand transform Journal of Mathematical Science, 2012, Volume 408 Join work with Michael Hinz and Alexander Teplyaev.

52 Part III: Intrinsic metrics on fractals Idea To deal with these problems by developing a differential geometry for Dirichlet spaces and hence fractals. Differential forms on the Sierpinski gasket Fabio Cipriani, Jean-Luc Sauvageot Marius Ionescu, Luke Rogers, Alexander Teplyaev Vector analysis on Dirichlet Spaces Michael Hinz, Michael Röckner, Alexander Teplyaev

53 Dirichlet Forms Let (X, d) be a locally compact separable metric space with a nonnegative Radon measure µ on X, with µ(u) > 0 for all non-empty U X. A pair (E, F) is called a symmetric Dirichlet Form on L 2 (X, µ) if (DF1) E : F F R is a nonnegative definite bilinear form on a dense subspace F L 2 (X, µ). (DF2) (E, F) is closed i.e. (F, E 1 ) is a Hilbert space, where E 1 (f, g) = E(f, g) + f, g L2. (DF3) Markov property u F implies that ũ = max {min {u, 0}, 1} F and E(ũ) E(u). (DF4) C := C c (X) F is uniformly dense in compactly supported continuous functions C c (X) and dense in F with respect to the Hilbert space norm E 1 (f) 1/2.

54 Dirichlet Algebra and Energy Measures Dirichlet algebra C := C c (X) F is an R-algebra of bounded functions, since E(fg) 1/2 f L (X,µ) E(g)1/2 + g L (X,µ) E(f)1/2, f, g C, Energy Measures f C we may define a nonnegative Radon measure Γ(f) on X by ϕ dγ(f) = E(ϕf, f) 1 2 E(ϕ, f 2 ), ϕ C.

55 Energy dominant measures A nonnegative Radon measure m on X with m(u) > 0 for any nonempty open U X is called energy dominant for (E, F) if all energy measures Γ(f), f C, are absolutely continuous with respect to m. Note that the original measure µ is energy dominant for (E, F) if and only if (E, F) admits a carré du champ, a la Bouleau and Hirsch.

56 coordinate sequence for (E, F): A sequence of functions (f n ) n=1 C if the span of {f n} n=1 is E 1-dense in F and Γ(f n ) m for any n. Lemma Let (E, F) be a regular symmetric Dirichlet form on L 2 (X, µ). Then there exist a finite energy dominant measure m 0 and a point separating coordinate sequence (f n ) n for (E, F) with respect to m 0. For any summable sequence (a n ) n (0, 1) the sum of measures m 0 := n: E(f n)>0 a n Γ(f n ) (3) is energy dominant (e.g. Kusuoka measure on the Sierpinski Gasket)

57 Let (E, F) be a strongly local Dirichlet form on L 2 (X, µ). Then we can define (Radon) energy measures Γ(f) for functions f from F loc = {f L 2,loc (X, µ) for any K X compact there exists some u F with u K = f K µ-a.e.} by setting Γ(f) := Γ(u), seen as a measure on K, if u F loc is such that u K = f K µ-a.e.

58 Now let m be an energy dominant measure for (E, F). For simplicity we use the symbol Γ(f) also to denote the density dγ(f)/dm. Set A := {f F loc C(X) Γ(f) L (X, m)}. (4) The intrinsic metric or Carnot-Caratheodory metric induced by (E, F) and m is defined by d Γ,m (x, y) := sup {f(x) f(y) : f A with Γ(f) m} (5)

59 Let C 0 (X) denote the space of continuous functions on X that vanish at infinity and consider the space A 0 := {f F loc C 0 (X) Γ(f) L (X, m)}. (6) Set A 1 0 := {f A 0 : Γ(f) m}. Theorem Let (E, F) be a strongly local Dirichlet form on L 2 (X, µ), let m be an energy dominant measure for (E, F) and assume there exists a point separating coordinate sequence for (E, F) with respect to m. Then A 1 0 is compact in C 0(X) if and only if d Γ,m induces the original topology.

60 If the reference measure µ itself is energy dominant, by Stollmann/Sturm: If d and d Γ,µ induce the same topology, then (X, d Γ,µ ) is a length space. However, a look at the proofs and their background reveals that this conclusion remains valid for any energy dominant measure m, provided a point separating coordinate sequence exists. Corollary Let (E, F) be a strongly local Dirichlet form on L 2 (X, µ), let m be an energy dominant measure for (E, F), and assume there exists a point separating coordinate sequence for (E, F) with respect to m. If A 1 0 is compact in C 0(X), then the metric space (X, d Γ,m ) is a length space.

61 Resistance form For any set X, a pair (E, F) is called a resistance form on X if (RF1) E : F F R is a nonnegative definite symmetric bilinear form on a vector space F of real valued functions on X, and E(u) = 0 if and only if u is constant on X, (RF2) (F/, E) is a Hilbert space; here is the equivalence relation on F given by u v if u v is constant on X, (RF3) Markov property, E(ũ) E(u), (RF4) Resistance metric d R (x, y) := sup { u(x) u(y) 2 : u F, E(u) 1 } (RF5) F separates the points of X. Confer with Kigami 2012 Memoir.

62 Theorem (Hinz K. Teplyaev) Suppose (E, F) is a local resistance form on X such that (X, d R ) is compact, let m be an energy dominant measure for (E, F), and assume there exists a coordinate sequence for (E, F) with respect to m. Then the topologies induced by d R and d Γ,m coincide, and (X, d Γ,m ) is a length space. Further, the space (X, d Γ,m ) is compact and therefore complete and geodesic.

63 Differential forms on Dirichlet spaces Let X be a locally compact second countable Hausdorff space and m be a Radon measure on X with full support. Let (E, F) be a regular symmetric Dirichlet form on L 2 (X, m) and write again C := C 0 (X) F. The space C is a normed space with f C := E 1 (f) 1/2 + sup f(x). x X

64 Differential forms on Dirichlet spaces We equip the space C C with a bilinear form, determined by a b, c d H = bd dγ(a, c). This bilinear form is nonnegative definite, hence it defines a seminorm on C C. X H: the Hilbert space obtained by factoring out zero seminorm elements and completing.

65 Differential forms on Dirichlet spaces We call H the space of differential 1-forms associated with (E, F). The space H can be made into a C-C-bimodule by setting a(b c) := (ab) c a (bc) and (b c)d := b (cd) and extending linearly. C acts on both sides by uniformly bounded operators.

66 Differential on Dirichlet spaces we can introduce a derivation operator by defining : C H by a := a 1. a 2 2E(a) and the Leibniz rule holds, (ab) = a b + b a, a, b C.

67 Co-Differential on Dirichlet spaces The operator extends to a closed unbounded linear operator from L 2 (X, m) into H with domain F. Let denote its adjoint, such that ω, g H = ω, g L2 (X,m) (7) Let C be the dual space of the normed space C. Then defines a bounded linear operator from H into C.

68 Dirac operators on Dirichlet spaces Define the Hilbert space H := L 2 (X, m) H with the natural scalar product (f, ω), (g, η) H := f, g L2 (X,m) + ω, η H.

69 Dirac operators on Dirichlet spaces Put dom D := F dom and define an unbounded linear operator D : H H by D(f, ω) := ( ω, f), (f, ω) dom D. To D we refer as the Dirac operator associated with (E, F). In matrix notation its definition reads ( ) 0 D =. 0 Lemma The operator (D, dom D) is self-adjoint on H.

70 Square of the Dirac operator The square of D is given by ( D 2 = 0 0 ). (L, dom L): the infinitesimal L 2 (X, m)-generator of (E, F). Then L =. Set dom 1 := {ω dom : ω F} and 1 ω :=, ω dom 1. Lemma The operator (D 2, dom L dom 1 ) is self-adjoint on H.

71 If we are in a situation where our fractal is topologically 1-dimensional, then D 2 is the Hodge Laplacian. (see Tep+Hinz)

72 We are particularly interested in the special case where the generator L of (E, F) has pure point spectrum. e.g. if the spectrum is discrete: there are an increasing sequence 0 < λ 1 λ 2... of nonzero eigenvalues λ i of L, with possibly infinite multiplicities taken into account, and an orthonormal basis {ϕ j } j=1 in L 2(X, m) of corresponding eigenfunctions such that Lf = λ j f, ϕ j L2 (X,µ) ϕ j, f dom L, (8) j=1 and zero itself may be an eigenvalue of infinite multiplicity.

73 Lemma If the generator L of (E, F) has pure point spectrum with spectral representation (8) then D admits the spectral representation on the orthogonal compliment of ker D, Dv = where j=1 λ 1/2 j v, v j H v j j=1 λ 1/2 j v, w j H w j, v dom D, (9) v j = 1 2 (ϕ j, λ 1/2 j ϕ j ) and w j = 1 2 (ϕ j, λ 1/2 j ϕ j ) In general, any eigenvalue of D may be of infinite multiplicity. ker D consists of the 0-eigenvalue and 1-forms which are orthogonal to the image of.

74 Lemma Assume that L has pure point spectrum with spectral representation (8). Then the operator (D 2, dom D 2 ) admits the spectral representation D 2 v = i=0,1 j=1 λ j v, v j,i H v j,i, v dom D 2. Where v j,0 = (φ j, 0) and v j,1 = (0, w j ). Again, any eigenfunction of D may be of infinite multiplicity.

75 Examples The Sierpinski gasket X = SG, equipped with the resistance metric and the natural self-similar normalized Hausdorff measure µ. Let (E, F) be the local regular Dirichlet form on L 2 (SG, µ), determined by the standard energy form on SG. It is known that the generator of (E, F) has discrete spectrum. This can also be observed for more general resistance forms on p.c.f. self-similar sets.

76 Examples Generators of local regular Dirichlet forms on generalized Sierpinski carpets, considered with the natural normalized Hausdorff measure, have pure point spectrum.

77 Examples Let X = R n and let (E, F) be the quadratic form associated with a Schrödinger operator H = + V. Under some conditions on the potential V (for instance continuity and nonnegativity) the associated form will be a Dirichlet form, and under further conditions on V (for instance unboundedness at infinity) the operator H will have discrete spectrum. This also applies to relativistic Schrödinger operators and, more generally, to Schrödinger operators associated with Lévy processes. As well as Magnetic Schödinger operators.

78 Appendix: Spectral Triples Definition A (possibly kernel degenerate) spectral triple for an involutive algebra A is a triple (A, H, D) where H is a Hilbert space and (D, dom D) a self-adjoint operator on H such that (i) there is a faithful -representation π : A L(H), (ii) there is a dense -subalgebra A 0 of A such that for all a A 0 the commutator [D, π(a)] is well defined as a bounded linear operator on H, (iii) the operator (1 + D) 1 is compact on (ker D).

79 Spectral Triples A 0 := {f C : Γ(f) L (X, µ)} (10) Theorem A := clos C0 (X)(A 0 ). (11) Let (E, F) be a regular symmetric Dirichlet form on L 2 (X, µ) and assume µ is energy dominant for (E, F). Then we have the following. (i) There is a faithful representation π : A L(H), (ii) For any a A 0 the commutator [D, π(a)] is a bounded linear operator on H, (iii) If the L 2 (X, µ)-generator L of (E, F) has discrete spectrum then (1 + D) 1 is compact on (ker D), and (A, H, D) is a spectral triple for A.

80 Connes distance d D (x, y) := sup {a(x) a(y) a A 0 is such that [D, a] 1} defines a metric in the wide sense on X, (a version of) the Connes distance. Theorem Let (E, F) be a strongly local Dirichlet form on L 2 (X, µ), let m be energy dominant for (E, F) and assume there exists a point separating coordinate sequence for (E, F) with respect to m. Let D, A 0 and A be as above. Then d D (x, y) := sup {a(x) a(y) a A 0 is such that [D, a] 1} is a metric in the wide sense on X and d D d Γ,µ. If X is compact then d D = d Γ,µ.

81 Gelfand transform We use the theory of Banach algebras and the Gelfand transform to to consider Dirichlet forms on measurable spaces where the topology is not defined in advanced. That is if we start off with a measure space (X, X, µ), and a non-negative bilinear form E with suitable domain, then we use the relationship between the Gelfand transform and Daniell-Stone integration theory to extend E to a closable form on a space which has X embedded.

82 Thank you!!!