Existence of resistance forms in some (non self-similar) fractal spaces Patricia Alonso Ruiz D. Kelleher, A. Teplyaev University of Ulm Cornell, 12 June 2014
Motivation X Fractal
Motivation X Fractal Laplacian
Motivation X Fractal Dirichlet form Laplacian
Motivation X Fractal Dirichlet form Laplacian resistance form
Motivation X Fractal Dirichlet form Laplacian resistance form X is self-similar: p.c.f. (Kigami), Sierpiński carpet (Kajino),... X not self-similar: homogeneous random p.c.f. (Hambly), homogeous random carpets (Hambly, Kumagai, Kusuoka,...), self-conformal (Strichartz, Freiberg,...), fractafolds (Strichartz), fractal fields (Hambly, Kumagai), homogeneous non self-similar carpets (Steinhurst)
Motivation X Fractal Dirichlet form Laplacian resistance form X is self-similar: p.c.f. (Kigami), Sierpiński carpet (Kajino),... X not self-similar: homogeneous random p.c.f. (Hambly), homogeous random carpets (Hambly, Kumagai, Kusuoka,...), self-conformal (Strichartz, Freiberg,...), fractafolds (Strichartz), fractal fields (Hambly, Kumagai), homogeneous non self-similar carpets (Steinhurst), fractal quantum graphs
Quantum graphs graph: (V, E, ) V vertices -finite or countable many- E edges : E V V orientation r : E [0, ) weight function
Quantum graphs graph: (V, E, ) V vertices -finite or countable many- E edges : E V V orientation r : E [0, ) weight function a d c b
Quantum graphs graph: (V, E, ) V vertices -finite or countable many- E edges : E V V orientation r : E [0, ) weight function a d c b
Quantum graphs graph: (V, E, ) V vertices -finite or countable many- E edges : E V V orientation r : E [0, ) weight function a d c b
Quantum graphs d Weighted graph: (V, E,, r) V vertices -finite or countable many- E edges : E V V orientation 1.5 a 1 2 c b 2.5 r : E [0, ) weight function
Quantum graphs Weighted graph: (V, E,, r) G a d b Metric graph: 1-d simplicial complex c
Quantum graphs Weighted graph: (V, E,, r) G a d b Metric graph: 1-d simplicial complex c Φ e : [0, r(e)] G smooth, e E Lebesgue measure on G
Quantum graphs Weighted graph: (V, E,, r) G a d b Metric graph: 1-d simplicial complex c Φ e : [0, r(e)] G smooth, e E Lebesgue measure on G
Quantum graphs Weighted graph: (V, E,, r) G a d b Metric graph: 1-d simplicial complex c Quantum graph: metric graph + operator (-2nd derivative on edges) vertex conditions
Quantum graphs Weighted graph: (V, E,, r) G a d b Metric graph: 1-d simplicial complex c Quantum graph: metric graph + operator (-2nd derivative on edges) vertex conditions L 2 (G) := L 2 ([0, r(e)], dx) e E H 1 (G) := H 1 ([0, r(e)], dx) e E
Fractal quantum graphs Definition: A separable compact connected and locally connected metric space (X, d) is called a fractal quantum graph (FQG) if (i) there exists {l k } k=1 R + and isometries φ k : [0, l k ] X, k 1 such that and (ii) the set is totally disconnected. φ k ([0, l k ]) = [0, l k ] k 1 φ k ((0, l k )) φ j ((0, l j )) =, k j, X \ φ k ((0, l k )) k=1
Resistance form For each n 0, define F n := {f C(X ) f φ k H 1 ([0, l k ]) k n, n f loc. const. in X \ φ k ((0, l k ))} and the bilinear form k=1 E n (f, g) := X f (x)g (x) dx f, g F n.
Resistance form For each n 0, define F n := {f C(X ) f φ k H 1 ([0, l k ]) k n, n f loc. const. in X \ φ k ((0, l k ))} and the bilinear form k=1 E n (f, g) := X f (x)g (x) dx = n lk k=1 0 (f φ k ) (g φ k ) dx f, g F n.
Let D n be a dense subset of n k=1 φ k ((0, l k )) and E n (f, f ) := inf{e n (g, g) g F n, g Dn f }, f l(d n ) := {f : D n R}. Proposition: {E n, l(d n )} n 1 is a compatible sequence (of resistance forms).
Resistance form Theorem: Let X be a FQG and D a countable dense subset of X. R (i) There exists a resistance form (E, F) on Ω := D, where { } f (x) f (y) 2 R(x, y) := sup f F E(f, f ) is the resistance metric associated to (E, F). (ii) Furthermore, if there exists a sequence {ε n } n 1 such that ε n 0 and ) n diam R (conn. comp. Ω \ φ k (0, l k ) < ε n n 1 k=1 then (E, F) is a resistance form on X. ( )
Dirichlet form and Laplacian Theorem 2: Property ( ) X is R-compact.
Dirichlet form and Laplacian Theorem 2: Property ( ) X is R-compact. (E, F) regular resistance form + µ locally finite and regular
Dirichlet form and Laplacian Theorem 2: Property ( ) X is R-compact. (E, F) regular resistance form + µ locally finite and regular (E, D) local and regular Dirichlet form on L 2 (X, µ)
Dirichlet form and Laplacian Theorem 2: Property ( ) X is R-compact. (E, F) regular resistance form + µ locally finite and regular (E, D) local and regular Dirichlet form on L 2 (X, µ) µ associated Laplacian on X
Example: Hanoi-type FQG 1. E n (f, g) := X f (x)g (x) dx f, g F n 2. E n (f, g) f, g l(d n ) 3. (E, F) on X
Example: Hanoi-type FQG (E, F) resistance form + µ β measure µβ Let r := 1 α 2 and s := 1 β 3 and N D/N (x) denote the eigenvalue counting function of N/D µ β. Then (i) N D/N (x) x 1 2, for 0 < rs < 1 9, (ii) N D/N (x) x 1 2 log x, for rs = 1 9, (iii) N D/N (x) x log 3 log(rs), for rs > 1 9.
Work in progress: generalization Let (X, d) be a compact connected metric space satisfying X = A i, I countable set, A i X open, i I for all i I, (X \ A i ) A i finite. Theorem: For such space (X, d), if (i) There exists (E i, F i ) resistance form on A i for each i I and the resistance metric R i is compatible with d. (ii) There exists M > 0 such that for any x, y X γ : [0, 1] X s.t. diam Ri (A i ) < M, ( ) i I γ A i then, there exists a resistance form (E, F) on X.
Thank you!