Vector Analysis on Fractals

Transcription

1 Vector Analysis on Fractals Daniel J. Kelleher Department of Mathematics Purdue University University Illinois Chicago Summer School Stochastic Analysis and Geometry Labor Day Weekend

2 Introduction: Dirichlet Forms We have the classical Dirichlet energy, on R n D(f) = f dx f H 1 (R n ) by H 1 (R n ), can be either seen as the closure of C0 (Rn ) with respect to the norm f 2 H 1 = f 2 + f 2 dx. Or you can think of H 1 (R n ) as the set of functions f L 2 (R) such that f L 2 (R),

3 Introduction: Dirichlet Forms Dirichlet s method: which was an early technique in the calculus of variationa. This was to solve Laplace equation { f = 0 in Ω f Ω = g for some prescribed boundary condition g : Ω R. And realizing that the solutions is the function u which minimizes D(f) such that f Ω = g.

4 Introduction: Dirichlet Forms A Dirichlet form E is a generalization of D, and has the following definition... Say (X, d) is a locally compact metric space, with Borel regular measure µ. We define (E, Dom E ), where Dom E L 2 (X) on L 2 (X, µ) if (DF1) E : Dom E Dom E R is a nonnegative definite bilinear form on a dense subspace F L 2 (X, µ). (DF2) (E, Dom E ) is closed i.e. (Dom E, E 1 ) is a Hilbert space, where E 1 (f, g) = E(f, g) + f, g L2. (DF3) Markov property u Dom E implies that ũ = max {min {u, 0}, 1} Dom E and E(ũ) E(u). (DF4) C := C c (X) Dom E is uniformly dense in compactly supported continuous functions C c (X) and dense in Dom E with respect to the Hilbert space norm E 1 (f) 1/2.

5 Introduction: Dirichlet Forms Dirichlet Forms Laplacians Infinitesimal Generator Markov Processes Hille Yosida Resolvents Semigroups transistion probabilities

6 Dirichlet Forms and the Hille Yosida Theorem there is a unique self adjoint operator A, with Dom A, such that E(f, g) = f, Ag µ = fag dµ. There is also a semi-group P t = e t such that u(x, t) = P t f, is the solution to the heat equation Au(x, t) = u (x, t) t u(x, 0) = f(x). Thus we can get A back from P t, by f P t f Af(x) = lim. t 0 t

7 Dirichlet Forms and Markov Processes Markov process, X t on the space, which has L as its infinitesimal generator, and that E x f(x t ) = P t f(x). In the classical case, X t is Brownian motion.

8 Graph Energy Say G = (V, E) is a graph, and we say that x y if x is connected to y, then if we have edge weights c x,y we define the graph energy on G with domain l(v ) = {f : V R}, by E(f) = x y c x,y (f(x) f(y)) 2.

9 Introduction: Self-Similar fractals Classical iterated function systems: for a subset of R n F i are contraction maps for i X K is the unique compact subset of R n with K = i X F i (K)

10 Harmonic Structures The idea is to find a self-similar Dirichlet form, that is there are {r i } i X such that E(f) = i X r 1 i E(f F i ).

11 Harmonic Structures Post-Critically Finite fractals are self-similar structures with the boundary between cells (and the inverse images under the F i ) is finite. In this case, Kigami proved that it was sufficient to prove the existence of compatible energies on approximating graphs.

12 Sierpinski Gasket The Sierpinsk gasket SG: approximating graphs G k with vertex sets V 0 V 1 V 2 SG. We have compatible graph energies E k on G n, E j (f) = inf { E k (g) g Vj = f }. Because of the symmetry of the Sierpinski gasket E j is the graph energy on V j scaled by 5 j.

13 Sierpinski Gasket Then there is a self-similar form E(f) = lim E j (f Vj ) Where Dom E = {f : SG R E(f) < }.

14 Resistance forms 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.

15 Resistance forms 1. All finite energy functions are (Hölder) continuous with respect to the resistance metric because d R (x, y) u(x) u(y) 2 E(u) 2. for any finite set V X, then the discrete energy exists. tr V E(f) = inf {E(g) g F, g V = f}

16 Resistance forms Typically, one considers resistance forms the way we considered the Sierpinski gasket Step 1 Find a nested sequence of sets of points V k which become dense in your set X, Step 2 Find an energies E k that are compatible in that tr Vj E k = E j. Step 3 Prove that the resistance metric is consistent with whatever preexisting structure you had.

17 Harmonic Structures Computation of Resistance Scaling of the Pillowcase and Fractalina Fractals Michael J. Ignatowich, Catherine E. Maloney, David J. Miller, Khrystyna Nechyporenko. We proved the existence of harmonic structures to Pillow and fractalina fractals.

18 Resistance forms on Sierpinski Gasket We still don t have a Dirichlet form, because we don t have a measure. The first natural choice is the self-similar measure, which assigns each of the three sub-copies of the Sierpinski gasket weight 1/3. Also nice because the Laplacian is just the scaled limit of the graph Laplacians associated V k.

19 What s not in the Domain of the Laplacian So what is the problem? No nice functions are in the domain of the Sierpinski gasket. We will get into the root cause of this in a little bit.

20 Gaps in the spectrum of 3N-gaskets Preprint with Nikhaar Gupta, Maxwell Margenot, Jason Marsh, Alexander Teplyaev. Letting be such that E(f, g) = f, g Laplacians of compatible energies are related by the Schur Compliment. This can be used to compute the spectrum of the Laplacian.

21 Spectrum of the Laplacian of 3N-gaskets T R(z) = N( z) ztn( z)(2t N (1 2z)+2U N 1 (1 2z)+1) U N 1( z) (z 1) zu N 1( z)(2t N (1 2z)+2U N 1 (1 2z)+1) if N is even if N is odd. (1) Where T k and U k are Chebyshev polynomials of the 1st and 2nd kind respectively.

22 Spectrum of the Laplacian of 3N-gaskets If = lim n c n n, then the spectrum { σ( ) = lim c n+1 R n (z) z σ( n ) } n

23 Gaps in the spectrum Bob Strichartz and Co. showed that if there are asymptotic gaps in the spectrum of the Laplacian, then the Fourier coefficients of functions converge uniformly. Gaps in the spectrum is related to having a totally disconnected (real) Julia set of R.

24 Heat Kernel Estimates The heat kernel p(t, x, y) : R + K K R of the operator with respect to µ is the integral kernel such that, if f L 2 (K) and u : R + K R is defined P t f(x) = p(t, x, y)f(y) dµ(y)

25 Heat Kernel Estimates Using the theory from Kigami s 2012 memoir. (or using Barlow s 1996 book) The Laplacian on K has a symmetric heat kernel p(t, x, y) : R + K K R which satisfies the following estimates ( p(t, x, y) c 1 t d H/d w exp c 2 (d (x, y) dw /t) 1/(dw 1)) (2) and ( p(t, x, y) c 3 t d H/d w exp c 4 (d (x, y) dw /t) 1/(dw 1)). Here d H = log(3n)/ log(n + 1) is the Hausdorff dimension with respect to d, d w = α + d H = (log(2n + 3) + log(n))/ log(n + 1) where α = log ρ/ log(n + 1), ρ = c/(3n) = 1 + 2N/3, and c 1,c 2,c 3,c 4 are positive real constants.

26 Heat Kernel Estimates ( p(t, x, y) c 1 t d H/d w exp c 2 (d (x, y) dw /t) 1/(dw 1)) (3) and ( p(t, x, y) c 3 t d H/d w exp c 4 (d (x, y) dw /t) 1/(dw 1)). In the classical case, we have Guassian d H is the dimension of the space, and d w = 2. These are called sub-gaussian, because d w < 2.

27 Further work: Localized Eigenfunctions Teplyaev, proved that the Sierpinski gasket has an eigenbasis of eigenfunctions which are supported on individual subcells of the Sierpinski Gasket. It follows from our work (not explicitly) that there are such localized eigenfunctions on the other 3N-gaskets.

28 Energy Measures and the Harmonic Gasket Jun Kigami,Measurable Riemannian Geometry on the Sierpinski Gasket Naotaka Kajino, Analysis of the measurable Riemannian structure on the Sierpinski gasket

29 Energy Measures What we want, is something to take the place of f 2, or it s polarization f g. There are some fairly prominent ways of doing thins 1. Bakry-Emery Upper gradients

30 Local energy forms A Dirichlet form (E, Dom E ) is strongly local if, for u, v Dom E such that u is constant on the support of v, then E(u, v) = 0.

31 Beurlung Deny decomposition For a regular Dirichlet form, there is a strongly local Dirichlet form (E c, Dom E ), and measure κ on X and J on X X \ {(x, x)} such that E(u, v) = E c (u, v) + (u(x) u(y))(v(x) v(y)) J(dx, dy) + X X\ X u(x)v(x) κ(dx) J(dx, dy) is the jump measure and κ is the killing measure. The names of these measures correspond to parts of the Markov process.

32 Energy Measures we see that for f C = C b (X) Dom E φ E(fφ, f) 1 2 E(φ, f 2 ) is a bounded linear function on C 0 (X), and thus there is a measure, Γ(f) such that φdγ(f) = E(fφ, f) 1 2 E(φ, f 2 )

33 Energy Measures The nice part is, if E is strongly local, then Γ behaves like a bilinear differential, if F : R n R, then and f(x) = (f 1 (x),..., f n (x)) then Γ(F f, h) = F x i Γ(f i, h).

34 Carré du Champ Notice that for f Dom A with f 2 Dom A, then φdγ(f) = φ( faf Af 2 ) dµ. And thus dγ(f)/dµ = faf Af 2 is the Carré du champ of Barkry Emery.

35 But of course this is too much to ask for of the Sierpinski gasket. However, we can weaken things and ask simply that Γ(f) µ for a good family of f at least. If this is the case, we say that the Dirichlet form admits a Carré du champ and shall say that µ is energy dominant. In fact things on the Sierpinski gasket are worse, if we take the self-similar measure... in fact all of the measures Γ(f) are singular with respect to the self similar measure µ.

36 Harmonic Sierpinski Gasket We introduce the Harmonic embedding of the Sierpinski gasket: if we take h 1 and h 2 to be the harmonic functions which have values (0, 1, 1/2) and (0, 0, 3/2) on V 0 respectively, then we have a map Φ : SG R 2 by Φ(x) = (h 1 (x), h 2 (x)) The we define the Kusuoka measure ν = Γ(h 1 ) + Γ(h 2 ).

37 Harmonic Sierpinski Gasket The we define the Kusuoka measure ν = Γ(h 1 ) + Γ(h 2 ). The collection of functions Φ f such that f C 1 (R 2 ) are dense in the Dom E. Further there is a matrix-valued function Z x, such that E(Φ f) = f(x), Z x f(x) dν(x).

38 Differential forms on fractals Idea: to deal with these problems by developing a differential geometry for Dirichlet spaces and hence fractals based on Differential forms on the Sierpinski gasket and other papers by Cipriani Sauvageot Derivations and Dirichlet forms on fractals by Ionescu Rogers Teplyaev, JFA 2012 Vector analysis on Dirichlet Spaces by Hinz Röckner Teplyaev, SPA 2013

39 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

40 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.

41 Differential forms on Dirichlet spaces In the classical setting, this norm b 2 a 2 dµ Where µ is Lebesgue measure in the appropriate dimension. And, any simple tensor a b = d i=1 xi b a x i. Think of x i 1 as dx i,

42 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.

43 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.

44 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) (4) Let C be the dual space of the normed space C. Then defines a bounded linear operator from H into C.

45 PDE on fractals We can think of as something like a gradient or an exterior derivative. And think of as div or as the co-differential. This allows for a lot of new differential equations to but represented on fractals For instance, we now have a divergence form a( u) = 0

46 Magnetic Schrödinger operators Classically i u t = ( i A)2 u + V u becomes i u t = ( i a) ( i a)u + V u Where a C and V L (X, m).

47 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.

48 Dirac operator: Classical case The Dirac operator d + d On the space of all differential forms (with functions as 0-forms). (d + d ) 2 = dd + d d is the Laplacian extended to differential forms.

49 Dirac operator: Classical case The Hodge decomposition: Every differential n-form ω = dα + d β + γ where α is an n 1-form, β is an n + 1-form, and γ is a Harmonic n-form, i.e. (dd + d d)γ = 0.

50 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.

51 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.

52 If our fractal is topologically 1-dimensional, then D 2 is the Hodge Laplacian. (see Tep+Hinz)

53 Intrinsic metrics on fractals Metrics and Spectral Triples for Dirichlet and Resistance Spaces Journal of Noncommutative Geometry and Measures and Dirichlet forms under the Gelfand transform Journal of Mathematical Science, 2012, Volume 408

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 Now let m be an energy dominant measure for (E, F). For simplicity we use Γ(f) also to denote the density dγ(f)/dm. Set A := {f F loc C(X) Γ(f) L (X, m)}. (5) 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) 1} (6)

56 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)}. (7) Set A 1 0 := {f A 0 : Γ(f) 1}. 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.

57 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.

58 Compare with Classical Cases On a Reimannian Manifold, (M m, g), Intrinsic Metric L(γ) = b a g( γ, γ) dt d(x, y) = inf {L(γ) γ(a) = x, γ(b) = y} Same as inf {f(x) f(y) g( f, f) 1}

59 Theorem 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.

60 Harmonic Sierpinski Gasket As an example, any two points between in Φ(V n ), for any n, can be connected by a piecewise C 1 curve in Φ(SG), which minimizes such length. The length metric ρ is the same as d Γν.

61 Harmonic Sierpinski Gasket With respect to ρ and ν, the heat kernel on SG actually has Gaussian bounds. c L exp( ρ(x, y) 2 /c L t) V t (x) p ν (t, x, y) c u (1 + ρ(x, y) 2 /t) κ/2 (exp( ρ(x, y) 2 /4t) V t (x)1/2 V t (y)1/2

62 Problems with curvature Bakry-Emery CDE s: These curvature dimension inequalities related Γ(f) to Γ 2 (f) := 2AΓ(f) Γ(f, Af). While, we may know that Γ(f) is a function we don t necessarily know where it lies. Optimal Transport: Kajino showed that the Harmonic SG cannot satisfied the curvature dimension bounds from optimal transport.

63 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).

64 Spectral Triples A 0 := {f C : Γ(f) L (X, µ)} (8) A := clos C0 (X)(A 0 ) (9) π(a)(f, ω) := (af, aω), for a A (10) Theorem Let (E, F) be a regular symmetric Dirichlet form on L 2 (X, µ) and assume µ is energy dominant for (E, F). Then (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.

65 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.

66 Gelfand transform We can 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.

67 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. 3. Barlow Bass started proof of existence in 89, and uniqueness wasn t shown until 2010.

68 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 also Antoni Brzoska, Diwakar Raisingh, and Gabe Khan

69 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.

70 Start off with a simplex

71 Add midpoints to simplexes

72 Add edges to simpleces

73 ng B 2 (T 0 ) (left) and the graph isomorphism to G 2

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

75 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. (11) If i is even (j is still i + 1 mod 6), then xi1v xj1v and xi2v xj2v. (12) This produces a fractal with Cantor boundaries and hexagonal symmetries. We call it the hexacarpet.

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

77 (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)

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

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

80 Higher dimensions

81 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.

82 Eigenfunction Pictures - Level 4

83 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

84 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.

85 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.

86 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.

87 Heat kernel estimates Barlow and Bass used a probabilistic interpretation, 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 ) There may be some logarithmic corrections?

88 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?

89 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.

90

91 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. Topologically we still have the same Euclidean triangle! 2. The Hausdorff (and self-similarity) dimension of this object is log 2 (6) 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.

92 Non-p.c.f. structures Barlow and Bass proved the existence of a diffusion process on the Sierpinski Carpet. This Required two main ingredients 1. Resistance estimates (Barlow Bass 1990) 2. Harnack inequalities (Barlow Bass 1989,1999)

93 Resistance estimates The Barlow Bass resistance estimates use a weak form of duality between two graph approximations of the Carpet. The equivalent of these graphs in our case are the Hexacarpet and barycentric subdivision. If ρ is the resistance scaling factor for the Hexacarpet and ρ T is the resistance scaling of barycentric subdivisions, then ρ = 1/ρ T.

94 Research in progress Theorem There are limits 1 log ρ = lim n n log R n and log ρ T 1 = lim n n log RT n that satify estimates 5 4 ρ 3 2 and 2 3 ρt 4 5

95 Uniform Harnack Inequalities & Resistance estimates

96 Thank you!!!