5th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals June 11-15, 2014

Transcription

1 5th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals June 11-15, 2014 Speakers include: Eric Akkermans (Technion), Christoph Bandt (Greifswald), Jean Bellissard (GaTech), David Croydon (Warwick), Eva Curry (Acadia), Marcel Filoche (Ecole Polytechnique), Uta Freiberg (Stuttgart), Michael Hinz (Bielefeld), Jiaxin Hu (Tsinghua), Marius Ionescu (Colgate and UMD), Tommaso Isola (Roma Tor Vergata), Palle Jorgensen (Iowa), Jun Kigami (Kyoto), Takashi Kumagai (Kyoto), Michel Lapidus (California Riverside), Ka-Sing Lau (Hong Kong), Daniel Lenz (Jena), Sze-Man Ngai (Georgia Southern), Anders Oberg (Uppsala), Kasso Okoudjou (Maryland), Roberto Peirone (Univ. Roma Tor Vergata), Conrad Plaut (Tennessee), Hua Qiu (Nanjing), Martin Reuter (Mainz), Huojun Ruan (Zhejiang), Nageswari Shanmugalingam (Cincinnati), Yang Wang (Michigan State), Martina Zaehle (Jena) Organizers: Robert Strichartz (Chair, Cornell), Luke Rogers, Alexander Teplyaev (UConn) Partially supported by NSF grant DMS Students and junior researchers from underrepresented groups in STEM are particularly encouraged to apply for travel funding. fractals/

2 Previous Meetings Fractals 4 September 10 13, 2011 Fractals 3 June 11 15, 2008 Fractals 2 May 31 June 4, 2005 Fractals 1 Conference on Analysis and Probability on Fractals at Cornell University June 16 20, 2002

3 Intro Part I: From self-similar groups From Self-Similar Structures to Self-Similar Groups International Journal of Algebra and Computation Vol. 22, No. 07 (2012) by Dan Kelleher, Ben Steinhurst and Mike Wong. Idea: determine the relationship between dynamical systems arising from self-similar groups and self-similar fractals.

4 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

5 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

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 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

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 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.

8 Hanoi Towers Group and Sierpiński Gasket (Grigorchuk and Sunik) 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

9 Hanoi Towers Group and Sierpiński Gasket (Grigorchuk and Sunik) 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

10 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

11 When Does a Limit Space have a P.C.F. Self-Similar Structure? 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 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

12 From Kigami to Nekrashevych Theorem A Kigami p.c.f. self similar structure L = (K, X, {F i } i X ) is the limit space of a space of a Nekrashevych p.c.f. self-similar group 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.

13 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: (1, 1) (0, 1) g 2 g 1 1 (0, 0) (1, 0) Figure: The generators of G L Compare with the Grigorchuk group

14 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!

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

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

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

18 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.

19 Intro 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 also Antoni Brzoska, Diwakar Raisingh, and Gabe Khan

20 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.

21 Start off with a simplex

22 Add midpoints to simplexes

23 Add edges to simpleces

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

25 This A.Teplyaev, is what M.Hinz, the D.Kelleher 4th level approximation looks like.

26 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.

27 A.Teplyaev, M.Hinz, Figure 6.1. D.Kelleher Two-dimensional eigenfunction Intrinsic Metrics coordinates on Fractals (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)

28 A.Teplyaev, Figure M.Hinz, 6.2. D.Kelleher Three-dimensional Intrinsic eigenfunction Metrics coordinates on Fractals (a) (ϕ2, ϕ3, ϕ4) (b) (ϕ2, ϕ3, ϕ5) (c) (ϕ2, ϕ3, ϕ6) (d) (ϕ2, ϕ3, ϕ7) (e) (ϕ2, ϕ4, ϕ5) (f) (ϕ2, ϕ4, ϕ6) (g) (ϕ2, ϕ5, ϕ6) (h) (ϕ3, ϕ5, ϕ6) (i) (ϕ4, ϕ5, ϕ6)

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

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

31 Higher dimensions

32 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.

33 Eigenfunction Pictures - Level 4

34 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

35 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.

36 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.

37 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.

38 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 )

39 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...

40 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?

41 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.

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

43 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.

44 research in progress Theorem There are limits 1 ρ = lim n n log R n and ρ T 1 = lim n n log RT n that satify estimates 5 4 ρ 3 2 and 2 3 ρt 4 5 Remark: for similar questions on Euclidean [0, 1] d or on generalized Sierpinski carpets, one has ρ = ρ T