Line integrals of 1-forms on the Sierpinski gasket

Transcription

1 Line integrals of 1-forms on the Sierpinski gasket Università di Roma Tor Vergata - in collaboration with F. Cipriani, D. Guido, J-L. Sauvageot Cambridge, 27th July 2010 Line integrals of

2 Outline The Sierpinski gasket 1 The Sierpinski gasket 2 3 Line integrals of

3 Outline The Sierpinski gasket 1 The Sierpinski gasket 2 3 Line integrals of

4 Definition The Sierpinski gasket T R 2 equilateral triangle, side length 1, vertices V 0 := {v 1, v 2, v 3 } w i : x R (x v i) + v i R 2, i = 1, 2, 3 Then there is a unique compact K R 2 s.t. K = W (K ) 3 i=1 w i(k ), also obtained as lim n W n (T ). K is called Sierpinski gasket.... Line integrals of

5 Definitions The Sierpinski gasket K Σ Similarities: w σ = w σ1... w σn, σ = n Cells: K σ = w σ (K ) Then K = σ =n K σ Edges: E = n E n, E 0 = {e 1, e 2, e 3 }, E n = {w σ (e), σ = n, e E 0 }. o(e) = origin of e, t(e) = terminus of e Lacunas: l σ = w σ (l) Line integrals of

6 Definitions The Sierpinski gasket e 1 e 2 e 3 Similarities: w σ = w σ1... w σn, σ = n Cells: K σ = w σ (K ) Then K = σ =n K σ Edges: E = n E n, E 0 = {e 1, e 2, e 3 }, E n = {w σ (e), σ = n, e E 0 }. o(e) = origin of e, t(e) = terminus of e Lacunas: l σ = w σ (l) Line integrals of

7 Definitions The Sierpinski gasket Similarities: w σ = w σ1... w σn, σ = n Cells: K σ = w σ (K ) Then K = σ =n K σ Edges: E = n E n, E 0 = {e 1, e 2, e 3 }, E n = {w σ (e), σ = n, e E 0 }. o(e) = origin of e, t(e) = terminus of e Lacunas: l σ = w σ (l) Line integrals of

8 Topological and metric properties K is connected, locally connected, arcwise connected, but it is not semilocally simply connected [ = no universal cover]. α-dimensional Hausdorff measure of E R n : H α (E) := lim δ 0 Hausdorff dimension of E: inf E i A i diam A i δ (diam A i ) α i=1 d H (E) = inf{α > 0 : H α (E) = 0} = sup{α > 0 : H α (E) = + } Then d H (K ) = log 3 log 2 =: d, Hd (K ) (0, ), and H d (K ) = i=1 Hd (w 1 i (K )). Line integrals of

9 Dirichlet form Different way of approximating K : sequence of graphs (V n, E n )... Dirichlet (or energy) form on (V n, E n ): E n [f ] := x y (f (x) f (y))2. Then ( 5 nen 3) [f ] E[f ]. Set F := {f : E[f ] < }. Then F C(K ). Choose Borel regular prob. measure µ on K. Then (E, F) is local regular Dirichlet form on L 2 (K, µ). Therefore E(f, g) = (f, µ g), µ 0, compact resolvent. Line integrals of

10 Dirichlet form Different way of approximating K : sequence of graphs (V n, E n )... Dirichlet (or energy) form on (V n, E n ): E n [f ] := x y (f (x) f (y))2. Then ( 5 nen 3) [f ] E[f ]. Set F := {f : E[f ] < }. Then F C(K ). Choose Borel regular prob. measure µ on K. Then (E, F) is local regular Dirichlet form on L 2 (K, µ). Therefore E(f, g) = (f, µ g), µ 0, compact resolvent. Line integrals of

11 Laplacian for Bernoulli measure Different construction of µ for µ Bernoulli measure. for f : V n \ V 0 R set n f (x) := y x (f (x) f (y)) for f C(K ) set f (x) := 3 2 lim n 5 n n f (x), if limit dom( ) := {f C(K ) : f C(K )} F Obs. f dom( ) = f 2 dom( ). for f : V n R set ( f ν for f C(K ) set f ν (x) := lim n ( 5 3 Theorem (Gauss-Green) ) n (x) := y x (f (x) f (y)), x V 0 ) n ( f ) ν (x), if limit µ Bernoulli measure, f dom( ), g F. Then E(f, g) = K g f dµ + p V 0 g(p) f ν (p). n Line integrals of

12 Weyl-type asymptotics Classical Weyl asymptotic: Ω R n bdd connected open set. Define eigenvalue counting function: N(x) := λ x dim{f dom( ) : f = λf }. Then N(x) = cx n/2 (1 + o(1)), x. As for the gasket, G, a nonconstant 1 2 log 5-periodic function, s.t. N(x) = {G(log x 2 ) + o(1)}x ds/2, x, where d S := log 9 log 5, is the spectral exponent. Line integrals of

13 Harmonic functions for m N {0}, u : V m R,!f F s.t. f Vm = u, E[f ] = min{e[g] : g F, g Vm = u}. f is said m-harmonic. Theorem (weak maximum principle) f m-harmonic, σ multiindex, σ m, x w σ (K ). Then min wσ(v 0 ) f f (x) max wσ(v 0 ) f. Obs. Harmonic functions are dense in C(K ). Line integrals of

14 Outline The Sierpinski gasket 1 The Sierpinski gasket 2 3 Line integrals of

15 Universal 1-forms Want to construct differential 1-forms on K. d : g F 1 g g 1 F F Ω 1 (F) the F-bimodule generated by {fdg : f, g F}. It s called bimodule of universal 1-forms. Actions of F: { h.(fdg) = (hf )dg, h F (fdg).h = fd(gh) (fg)dh g(e) := g(t(e)) g(o(e)), g F, e E fdg Ω 1 (F). Line integrals of

16 1-forms The Sierpinski gasket Define (fdg, fdg) = lim n ( 5 3 bilinearity, fdg = lim n e e 1 E n,e 1 e ) n e E n f (o(e)) 2 g(e) 2, extended by f (o(e 1 )) g(e 1 ), extended by linearity. Then the limits above are well defined and finite for any ω Ω 1 (F), if e ω = 0 for any e E, then ω = 0. Definition (1-forms) Set Ω = Ω 1 (F)/, where ω 0 if e ω = 0 for any e. Line integrals of

17 n-exact 1-forms There exists a map {{f σ } σ =n, E Kσ [f σ ] < } ω Ω with ω Kσ = df σ. The form ω will be called n-exact. KΣ 0 σ Σ, let dz σ attain the min{ ω : ω Ω, is n + 1-exact, l σ ω = 1}. The form dz σ is zero on Kσ c, and is given by the (harmonic) function z σ on K σ. Then the set {dz σ } σ Σ is an orthogonal system, with dz σ 2 = 5/6(5/3) σ, the dz σ s are co-closed: (df, dz σ ) = 0, f F. Obs. Since K is topologically 1-dimensional, any 1-form is closed, hence we say that dz σ is a harmonic 1-form. Line integrals of

18 Hodge decomposition ω Ω!{k σ } σ Σ s.t., setting ω 0 = σ k σdz σ, we have N(k σ ) = sup n (5/3) n σ =n k σ <. ω 0 Ω, ω 0 2 = 5/6 σ (5/3) σ k σ 2 <, ω ω 0 is exact, i.e. U 1 F s.t. ω = du 1 + ω 0. Therefore ω = 0 = ω 0, i.e. Ω is a pre-hilbert space, Hodge decomposition: any 1-form in Ω can be uniquely decomposed into an exact and a harmonic part. Line integrals of

19 Outline The Sierpinski gasket 1 The Sierpinski gasket 2 3 Line integrals of

20 Coverings with finitely generated homotopy Let T = convex hull(k ), then i n : K T n := σ =n w σ (T ), i n : π 1 (K ) π 1 (T n ) = free group with #{ σ < n} generators. Let T n be the universal covering of T n. Then there exists a covering K n of K such that the diagram K n Tn p n K T n commutes, and deck( K n ) = deck( T n ) = π 1 (T n ). Line integrals of

21 Coverings with finitely generated homotopy The family {( K n, p n )} is projective. Take projective limit ( K, p) := lim ( K n, p n ). Then, any path γ in K has a lifting γ in K, unique up to the starting point. Obs. deck( K ) = lim deck(t n ) = ˇπ 1 (K ), the first Čech homotopy group of K, and ˇπ 1 (K ) π 1 (K ). Line integrals of

22 Abelian coverings A smaller covering. Set L n := K n /[deck( K n ), deck( K n )]. p n K n r n L n q n K Set ( L, q) = lim ( L n, q n ). Then deck( L) = lim deck( L n ), Then deck( L n ) = deck( K n )/[deck( K n ), deck( K n )] is a free abelian group with #{ σ < n} generators, {( L n, q n )} is a projective family. any path γ in K has a lifting γ in L, unique up to the starting point. Line integrals of

23 Affine functions. Potentials of n-exact 1-forms Say f C( K n ) is deck( K n )-affine if ϕ hom(deck( K n ), (C, +)) s.t. f (gx) f (x) = ϕ(g), x K n, g deck( K n ). Then Any n-exact 1-form ω has a unique (up to an additive constant) deck( K n )-affine potential U, for which e ω = U(e) := U(t(ẽ)) ( ) U(o(ẽ)), ẽ a lifting of e to K n 5 n ω 2 = E[U] = lim U(e) 2. n 3 e E n We denote by z σ the potential of dz σ. Line integrals of

24 Potentials of n-exact 1-forms live on abelian coverings Let ω, U, ϕ be as above. Since (C, +) is abelian, ϕ vanishes on commutators. Therefore, the potential U is a deck( L n )-affine function on L n. Obs. the projective limit topology on L is generated by {z σ : σ Σ}, any deck( L)-affine function on L is the lifting of a deck( L n )-affine function on L n, for some n. Line integrals of

25 Restricting the covering, pseudometrics ( ) 3 n sup a σ, so 5 σ =n a σ k σ N (a σ )N(k σ ), Set N (a σ ) = n σ and define d(x, y) = N ( z σ (y) z σ (x) ), x, y L. Then d is a pseudometric on L, namely a metric which is allowed to be infinite. The d-topology is finer than the projective limit topology. For any g deck( L), l(g) = d(x, gx) does not depend on x, and Γ = {g deck( L) : l(g) < } is a subgroup of deck( L). Γ acts on any d-component of L, namely on any subset of L consisting of points having finite mutual distance. Line integrals of

26 Integration on paths Any 1-form ω Ω has a Γ-affine potential U on any d-component of L. If ω = du 1 + σ k σdz σ, then U = U 0 + U 1, with U 0 = σ k σz σ. Such a sum converges uniformly on compact sets to a Γ-affine, d-continuous function on any d-component of L. Indeed, U 0 (x) U 0 (y) N(k σ )d(x, y). For any path γ in K, set l(γ) = d( γ(1), γ(0)). Then, if l(γ) <, and U is the potential of ω Ω, ω = U( γ(1)) U( γ(0)). γ Line integrals of