Noncommutative Geometry and Potential Theory on the Sierpinski - - PowerPoint PPT Presentation

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Noncommutative Geometry and Potential Theory on the Sierpinski Gasket

Fabio Cipriani

Dipartimento di Matematica Politecnico di Milano

• Joint works with D. Guido, T. Isola, J.-L. Sauvageot
• Neapolitan workshop on Noncommutative Geometry, Napoli, 20-22 September 2012

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Noncommutative Geometry underlying Dirichlet forms on singular spaces Usually one uses tools of NCG to build up hamiltonian on singular spaces NCG → Energy functionals                    Ground States; Algebraic QFT; Quantum Hall Effect; Quasi crystals; Standard Model; Action Principle; Heat equations on foliations; reversing the point of view, our goal is to analyze the NCG structures underlying Energy functionals Energy functionals → NCG    Compact Quantum Groups (with Franz, Kula); Orbits of Dynamical systems (with Mauri); Fractals (with Guido, Isola, Sauvageot). Today we concentrate on a fractal set: Sierpinski gasket K where the C∗-algebra of observables is commutative = C(K) Energy functionals = Dirichlet forms: a generalized Dirichlet integral on K

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Sierpinski gasket K ⊂ C: self-similar compact set vertices of an equilateral triangle {p1, p2, p3} contractions Fi : C → C Fi(z) := (z + pi)/2 K ⊂ C is uniquely determined by K = F1(K) ∪ F2(K) ∪ F3(K) as the fixed point of a contraction of the Hausdorff distance on compact subsets of C:

Duomo di Amalfi: Chiostro, sec. XIII

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Topological, geometric and analytic features of the Sierpinski gasket K is not a manifold K does not admit a universal cover the Hausdorff dimension is non integer 1 < dH < 2 Volume and Energy are distributed singularly on K Volume Zeta functions exhibits complex dimensions We will construct families of Spectral Triples on K reproducing Hausdorff volume measure µH Hausdorff dimension dH Euclidean geodesic metric as Connes’ distance Energy functional Energy dimension = Volume dimension

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Self-similar volume measures and their Hausdorff dimensions The natural volume measures on K are the self-similar ones for some fixed (α1, α2, α3) ∈ (0, 1)3 such that 3

i=1 αi = 1

• K

f dµ =

3

• i=1

αi

• K

(f ◦ Fi) dµ f ∈ C(K) when αi = 1

3 for all i = 1, 2, 3 then µ is the normalized Hausdorff measure

associated to the restriction of the Euclidean metric on K Hausdorff dimension dH = ln 3 ln 2 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Harmonic structure word spaces:

• 0 := ∅ ,
• m := {1, 2, 3}m ,

:=

m≥0

• m

length of a word σ ∈

m:

|σ| := m iterated contractions: Fσ := Fi|σ| ◦ . . . Fi1 if σ = (i1, . . . , i|σ|) vertices sets: V∅ := {p1, p2, p3} , Vm :=

|σ|=m Fσ(V0)

consider the quadratic form E0 : C(V0) → [0, +∞) of the Laplacian on V0 E0[a] := |a(p1) − a(p2)|2 + |a(p2) − a(p3)|2 + |a(p3) − a(p1)|2

• Theorem. (Kigami 1986)

The sequence of quadratic forms on C(Vm) defined by Em[a] :=

• |σ|=m

5 3 m E0[a ◦ Fσ] a ∈ C(Vm) is an harmonic structure in the sense that Em[a] = min{Em+1[b] : b|Vm = a} a ∈ C(Vm) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Dirichlet form, Laplacian and Spectral Dimension Theorem 1. (Kigami 1986) The quadratic form E : C(K) → [0, +∞] defined by E[a] := lim

m→+∞ Em[a|Vm]

a ∈ C(K) is a Dirichlet form, i.e. a l.s.c. quadratic form such that E[a ∧ 1] ≤ E[a] a ∈ C(K), which is self-similar in the sense that E[a] = 5 3

3

• i=1

E[a ◦ Fi] a ∈ C(K) . It is closed in L2(K, µ) and the associated self-adjoint (Laplacian) operator ∆µ has discrete spectrum with spectral dimension dS =

ln 9 ln 5/3 = dH:

Weyl ′s asymptotics ♯{eigenvalues of ´µ ≤ λ} ≍ λdS/2 λ → +∞ .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Volume and Energy measures Volume can be reconstructed as residue of Zeta functionals of the Laplacian

• Theorem. (Kigami-Lapidus 2001)

The self-similar volume measure µ with weights αi = 1/3 can be reconstructed as

• K

f dµ = TraceDix(Mf ◦ ∆−dS/2

µ

) = Ress=dSTrace(Mf ◦ ∆−s/2

µ

) Volume and Energy are distributed singularly on K

• Theorem. (Kusuoka 1989, Ben Bassat-Strichartz-Teplyaev 1999)

Energy measures on K defined by (Le Jan 1985)

• K

b dΓ(a) := E(a|ab) − 1 2E(a2|b) a, b ∈ F are singular with respect to all the self-similar volume measures on K. Energy/Volume singularity has a simple algebraic interpretation any sub-algebra contained in the domain of the Laplacian is trivial.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Differential calculus on K K is not a manifold so that a differential calculus has to be introduced in a unconventional way, more specifically there exists a natural differential calculus on K underlying the Dirichlet form (E, F).

• Theorem. (FC-Sauvageot 2003)

There exists a derivation (F, ∂, H), defined on the Dirichlet algebra F, with values in a Hilbertian C(K)-module H such that E[a] = ∂a2

H

a ∈ F . In other words, (F, ∂, H) is a differential square root of the Laplacian Hµ: Hµ = ∂∗ ◦ ∂ . Energy measures are represented as

• K b dΓ[a] = (∂a|b∂a)H

a ∈ F . Energy/Volume singularity has another algebraic interpretation: cyclic representations associated to vectors ∂a ∈ H are not weakly contained in the representation L2(K, µ).

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Construction of module and derivation Consider the universal derivation d : F → Ω1(F) over the algebra F the tangent C(K)-module H is the quotient/completion of the F-bimodule Ω1(F) under the inner product (a ⊗ b|c ⊗ d) = 1 2

• E(a|cdb∗) + E(abd∗|c) − E(bd∗|a∗c)
• the derivation ∂ is the quotient of the derivation d

The proof uses the following facts the algebra F is dense in C(K) positive linear maps on the commutative C∗-algebra C(K) are automatically completely positive efforts are required to prove that the quotient/completion of F is not only an F-bimodule but actually a C(K)-module coincidence of the left and right actions of C(K) on H is a consequence of the strong locality of the Dirichlet form a, b ∈ F , a constant on the support of b ⇒ E(a, b) = 0 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

A first Fredholm module on K

• Theorem. (FC-Sauvageot 2009)

Let P ∈ Proj(H) the projection onto the image Im∂ of the derivation above PH = Im∂ and let F := P − P⊥ the symmetry with respect to Im∂. Then (F, H) is an (ungraded) Fredholm module over C(K) (F, H) is 2-summable and densely defined on F Trace(|[F, a]|2) ≤ const. E[a] a ∈ F .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Proof By the Leibnitz rule for ∂ P⊥aP(∂b) = P⊥(∂(ab) − (∂a)b) = −P⊥((∂a)b) and P⊥aP(∂b) ≤ (∂a)b a, b ∈ F 0 < λ1 ≤ λ2 < . . . eigenvalues a1, a2, · · · ∈ L2(K, µ) normalized eigenfunctions ξk := λ−1/2

k

∂ak ∈ PH = Im ∂ complete orthonormal basis Since the Green function G(x, y) = ∞

k=1 λ−1 k ak(x)ak(y) is continuous on K

[F, a]2

L2 = 8P⊥aP2 L2 = 8 ∞

• k=1

λ−1

k P⊥aP(∂ak)2 H ≤ 8 ∞

• k=1

λ−1

k (∂a)ak2 H

= 8

∞

• k=1

λ−1

k

• K

a2

k dΓ(a) = 8

• K

G(x, x)Γ(a)(dx) ≤ 8G∞E[a] a = a∗ ∈ F . Since F is uniformly dense in C(K), we have that [F, a] is compact for all a ∈ C(K).

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Multipliers of the Dirichlet space To construct finer Fredholm modules over K, more sensible to the volume measure, we will use the following functions which lies at the core of the Potential Theory of the Dirichlet space (E, F).

• Definition. (Mokobodzki 2005)

A multiplier a ∈ C(K) is a continuous function such that b ∈ F ⇒ ab ∈ F . It is somewhat non trivial to exhibit non trivial multipliers: however, a tour de force in Potential Theory involving properties of finite energy measures and potentials of the Dirichlet space, allows to prove

• Theorem. (FC-Sauvageot arXiv:1207.3524v1)

The algebra M(F) of multipliers of the Dirichlet space F is dense in C(K); (I + ∆µ)−1b ∈ C(K) is a multiplier for all a ∈ L∞(K, µ); eigenfunctions are multipliers.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

A finer Fredholm module on K

• Theorem. (FC-Sauvageot arXiv:1207.3524v1)

Let F be the symmetry with respect to the graph G(∂∗) ⊂ L2(K, µ) ⊕ H of the divergence operator ∂∗. Then (F, L2(K, µ) ⊕ H) is a Fredholm module over C(K); it is (dS, ∞)-summable and densely defined on the multiplier algebra M(F) TraceDix(|[F, a]|dS) ≤ 4dS · adS

M(F) · TraceDix(I + ∆µ)−dS/2

a ∈ M(F) ; the energy functional EdS[a] := TraceDix(|[F, a]|dS) a ∈ M(F) is a self-similar conformal invariant in the sense that EdS[a] =

2

• i=0

EdS[a ◦ Fi] a ∈ M(F) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Dirac operator associated to the Dirichlet form

• Definition. (FC-Sauvageot arXiv:1207.3524v1)

The Dirac operator associated to the Dirichlet form (E, F) is defined as D = ∂∗ ∂

• acting on the C(K)-module L2(K, µH) ⊕ H.

Even if D does not gives rise to a Spectral Triple because [D, a] bounded ⇔ a = constant D helps to construct the previous Fredholm module because spectrum |D| = spectrum

• ∆µ, away from zero

[F, a] factorizes through |D| for all multipliers a ∈ M(F).

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Quasi-circles We will need to consider on the 1-torus T = {z ∈ C : |z| = 1} structures of quasi-circle associated to the following Dirichlet forms and their associated Spectral Triples for any α ∈ (0, 1).

• Lemma. Fractional Dirichlet forms on a circle (CGIS arXiv:0424604 25 Feb 2012)

Consider the Dirichlet form on L2(T) defined on the fractional Sobolev space Eα[a] :=

• T
• T

|a(z) − a(w)|2 |z − w|2α+1 dzdw Fα := {a ∈ L2(T) : Eα[a] < +∞} . Then Hα := L2(T × T) is a symmetric Hilbert C(K)-bimodule w.r.t. actions and involutions given by (aξ)(z, w) := a(z)ξ(z, w) , (ξa)(z, w) := ξ(z, w)a(w) , (J ξ)(z, w) := ξ(w, z) . The derivation ∂α : Fα → Hα associated to Eα is given by ∂α(a)(z, w) := a(z) − a(w) |z − w|α+1/2 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Proposition. Spectral Triples on a circle (CGIS arXiv:0424604 25 Feb 2012)

Consider on the Hilbert space Kα := L2(T × T) L2(T), the left C(T)-module structure resulting from the sum of those of L2(T × T) and L2(T) and the operator Dα :=

• ∂α

∂∗

α

• .

Then Aα := {a ∈ C(T) : supz∈T

• T

|a(z)−a(w)|2 |z−w|2α+1 < +∞} is a uniformly dense

subalgebra of C(T) and (Aα, Dα, Kα) is a densely defined Spectral Triple on C(T).

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Dirac operators on K. Identifying isometrically the main lacuna ℓ∅ of the gasket with the circle T, consider the Dirac operator (C(K), D∅, K∅) where K∅ := L2(ℓ∅ × ℓ∅) ⊕ L2(ℓ∅) D∅ := Dα the action of C(K) is given by restriction π∅(a)b := a|ℓ∅. Fix c > 1 and for σ ∈ consider the Dirac operators (C(K), πσ, Dσ, Kσ) where Kσ := K∅ Dσ := c|σ|Dα the action of C(K) is given by contraction/restriction πσ(a)b := (a ◦ Fσ)|ℓ∅ b. Finally, consider the Dirac operator (C(K), π, D, K) where K := ⊕σ∈Kσ π := ⊕σ∈πσ D := ⊕σ∈Dσ Notice that dim Ker D = +∞ and that D−1 will be defined to be zero on Ker D.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Volume functionals and their Spectral dimensions

• Theorem. (CGIS arXiv:0424604 25 Feb 2012)

The zeta function ZD of the Dirac operator (C(K), D, K), i.e. the meromorphic extension of the function C ∋ s → Trace(|D|−s) is given by ZD(s) = 4 1 − 3c−s z(αs) where z denotes the Riemann zeta function. The dimensional spectrum is given by Sdim = { 1 α} ∪ {ln 3 ln c

• 1 + 2πi

ln 3k

• : k ∈ Z}

and the abscissa of convergence is dD = max(α−1, ln 3

ln c ). When 1 < c < 3α there is a

simple pole in dD = ln 3

ln c and the residue of the meromorphic extension of

C ∋ s → Trace(f|D|−s) gives the Hausdorff measure of dimension d = ln 3

ln 2

TraceDix(f|D|−s) = Ress=dDTrace(f|D|−s) = 4d ln 3 z(d) (2π)d

• K

f dµ . Notice the complex dimensions and the independence of the residue Hausdorff measure upon c > 1.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Spectral Triples and Connes metrics on the Sierpinski gasket Spectral Triples on the Sierpinski Gasket constructed by [Guido-Isola 2005], [Christensen-Ivan-Schrohe arXiv:1002.3081v2] were able to reproduce the volume

• measure. The following one gives back volume measure, distance and energy.
• Theorem. (CGIS arXiv:0424604 25 Feb 2012)

(C(K), D, K) is a Spectral Triple for any 1 < c ≤ 2. In particular we have the commutator estimate for Lipschiz functions with respect to the Euclidean metric [D, a] ≤ (1/2)(1−α) (1 − α)1/2 sup

σ∈( c

2)σaLip(lσ) a ∈ Lip(K) . For c = 2 the Connes distance is bi-Lipschitz w.r.t. the geodesic distance on K induced by the Euclidean metric (1 − α)1/22(1−α)dgeo(x, y) ≤ dD(x, y) ≤ (1 + α)−1/22(3/2)3−αdgeo(x, y) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Energy functionals and their Spectral dimensions By the Spectral Triple it is possible to recover, in addition to dimension, volume measure and metric, also the energy form of K

• Theorem. (CGIS arXiv:0424604 25 Feb 2012)

Consider the Spectral Triple (C(K), D, K) for α ≤ α0 := ln 5

ln 4 − 1 2

ln 3

ln 2 − 1

• ∼ 0, 87

and assume a ∈ F. Then the abscissa of convergence of the energy functional C ∋ s → Trace(|[D, a]|2|D|−s) is δD := max(α−1, 2 − ln 5/3

ln c ).

If δD > α−1 then s = δD is a simple pole and the residue is proportional to the Dirichlet form Ress=δDTrace(|[D, a]|2|D|−s) = const. E[a] a ∈ F ; if δD = α−1 then s = δD is a pole of order 2 but its residue of order 2 is still proportional to the Dirichlet form. energy dimension and volume dimension differ: dD = δD.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Pairing with K-Theory Trying to construct a Fredholm module from the Dirac operator one may consider F := D|D|−1 to be the phase of the Dirac operator. ε0 := ⊕σεσ

0 where εσ 0 :=

I −I

• n Kσ = L2(ℓ∅ × ℓ∅) ⊕ L2(ℓ∅)

ε1 := ⊕σεσ

1 where εσ 1 :=

• −iVσ

−iV∗

σ

• n Kσ = L2(ℓ∅ × ℓ∅) ⊕ L2(ℓ∅)

where Vσ : L2(l∅) → L2(l∅ × l∅) is the partial isometry between Im (∂∗

σ ◦ ∂σ) ⊂ L2(ℓ∅) and Im (∂σ ◦ ∂∗ σ) ⊂ L2(ℓ∅ × ℓ∅).

However (C(K), F, K, ε0, ε1) is not a 1-graded Fredholm module because [F, π(a)] ∈ K(K) for all a ∈ C(K) F = F∗, ε0F + Fε0 = 0, but dim Ker F = +∞ and (F2 − I) / ∈ K(K) ε0 is unitary but ε1 is a partial isometry only ε0ε1 + ε1ε0 = 0, ε2

0 = I but ε2 1 = −I

[ε0, π(a)] = 0 but [ε1, π(a)] / ∈ K(K) in general

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Theorem. (CGIS arXiv:0424604 25 Feb 2012)

Consider the operator F1 := iε1F and the projection P :=

F1+F2

1

2

. Then [F1, π(a)] is compact for all a ∈ C(K) PaP is a Fredholm operator for all invertible u ∈ C(K) a nontrivial homomorphism on K1(K) is determined by u → Index PuP the nontrivial element of K1(K) =

σ∈ Z determined by F1 is (1, 1, . . . ):

Index PuσP = +1 σ ∈

• where uσ is the unitary associated to the lacuna ℓσ.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Proofs are based on the harmonic structure defining the Dirichlet form E a result

• A. Jonsson: A trace theorem for the Dirichlet form on the Sierpinski gasket,
• Math. Z. 250 (2005), no. 3, 599-609

by which the restriction to a lacuna ℓσ of a finite energy function a ∈ F belongs to the fractional Sobolev space Fα, if α < α0. These properties allow to develop integration of 1-forms ω ∈ H in the tangent module associated to (E, F) along paths γ ⊂ K characterize exact and locally exact 1-forms in terms of their periods around lacunas define a de Rham cohomology of 1-forms and prove a separating duality with the Cech homology group. represent the integrals of 1-forms around cycles in K by potentials, i.e. affine functions on the abelian universal projective covering L of K

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Elementary 1-forms and elementary paths Let (E, F) be the standard Dirichlet form on K and (F, ∂, H) the associated derivation with values in the tangent module H (whose elements are understood as square integrable 1-forms). exact 1-forms Ω1

e(K) := Im ∂ = {∂a ∈ H : a ∈ F}

elementary 1-forms Ω1(K) := {n

i=1 ai∂bi ∈ H : ai, bi ∈ F}

locally exact 1-forms Ω1

loc(K) are those ω ∈ Ω1(K) which admit a primitive

UA ∈ F on a suitable neighborhood A ⊂ K of a any point of K ω = ∂UA

• n

A ⊂ K . n-exact 1-forms are those ω ∈ Ω1(K) which are exact on any cell Cσ with |σ|=n an elementary path γ ⊂ K is a path which is a finite union of edges of K. If it is contained in a cell Cσ ⊂ K we say that γ has depth |σ| ∈ N.

• Lemma. (CGIS 2010)

A 1-form is locally exact if and only if it is n-exact form some n ∈ N.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Integration of elementary 1-forms along elementary paths

• Theorem. (CGIS 2010)

The integral of an elementary 1-form ω = n

i=1 ai∂bi ∈ Ω1(K) along an elementary

path γ ⊂ K is well defined as

• γ

ω :=

n

• i=1
• γ

aidbi where the

• γ aidbi is understood as a Lebesgue-Stieltjes integral.

Proof is a based on an embedding F Hα(γ) of the Dirichlet space into a fractional Sobolev space with α > 1/2. Potentials A continuous function U ∈ C(A) defined on subset A ⊂ K is a local potential

• n A of a 1-form ω ∈ Ω1(K) if for all elementary path γ ⊂ A
• γ

ω = U(γ(1)) − U(γ(0)) .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Proposition. (CGIS 2010)

Local potentials of a 1-form on A ⊂ K have finite energy on A the class of potentials U ∈ C(K) of an exact 1-form ω ∈ Ω1

e(K) coincides with

the class of its primitives U ∈ F on K

• γ

ω = U(γ(1)) − U(γ(0)) ⇔ ω = ∂U .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

A system of locally exact 1-forms associated with lacunas Cells and lacunas The lacuna ℓ∅ ⊂ K (depth 0) is defined as the boundary of the triangle K \ (F1(K) ∪ F2(K) ∪ F3(K)) the lacunas ℓσ (depth n) are defined as its successive contraction: ℓσ := Fσ(ℓ∅)

• Theorem. (CGIS 2010)

For any σ ∈ there exists only one (|σ| + 1)-exact 1-form dzσ ∈ Ω1

loc(K)

having minimal norm ωH among those with unit period on the lacuna ℓσ

• ℓσ

ω = 1 ∃ ωσ ∈ Lin {dzτ : |τ| ≤ |σ|} such that for all elementary paths γ ⊂ K

• γ

ωσ = winding number of γ around ℓσ .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Theorem. (a la de Rham) (CGIS 2010)

Any elementary 1-form ω ∈ Ω1(K) can be uniquely decomposed as ω = ∂U +

• σ

kσdzσ the convergence taking place in a topology where integrals are continuous the coefficients kσ only depend upon the periods

• ℓσ ω around lacunas

the form is locally exact ω ∈ Ω1

loc(K) iff the kσ’s are eventually zero

the form is exact ω ∈ Ω1

e(K) iff all the periods, hence all kσ’s, are zero

Cells and lacunas There exist elementary forms which are not closed: f0∂f1.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Theorem. (a la Hodge) (CGIS 2010)

The forms {dzσ : σ ∈ } ⊂ H are pairwise orthogonal Any 1-form ω ∈ H can be orthogonally decomposed as ω = ∂U +

• σ

kσdzσ the forms {dzσ.σ ∈ } are co-closed ∂∗(dzσ) = 0 so that ∂∗ω = ∂∗∂U = ∆U

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

• Theorem. (a de Rham cohomology Theorem) (CGIS 2010)

The pairing γ, ω =

• γ ω between elementary paths γ ⊂ K and locally exact forms

ω ∈ Ω1

loc(K) gives rise to a nondegenerate pairing between the Cech homology group

H1(K, R) and the cohomology group H1

dR(K) := Ω1 loc(K)

Ω1

e(K)

in which the classes of the co-closed forms {dzσ : σ ∈ } ⊂ H is a system of generators parameterized by lacunas.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Coverings Let T ⊂ C the closed triangle envelope of the gasket K and Tn :=

|σ|=n Fσ(T). The

embedding in : K → Tn give rise to regular coverings ( Kn, K) whose group of deck transformations can identified with π1(Tn). Its abelianization Γn can be identified with the homology group H1(Tn). Defining Ln := Kn/[Γn, Γn] one has that ( Ln, K) form coverings whose group of deck transformations can be identified with Γn.

• Theorem. () (CGIS 2010)

The family {( Ln, K) : n ≥ 0} is projective and the group Γ of deck transformations

• f its limit

L = lim← Ln can be identified with the Cech homology group of the gasket H1(K) ⋍ Γ := lim

← Γn .

The covering L has the unique lifting property.

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

Potentials of 1-forms

• Definition. Affine functions

A continuous function f ∈ C( L) is said to be affine if f(gx) = f(x) + φ(g) x ∈ L , g ∈ H1(K) for some continuous group homomorphisms φ : H1(K) → C. Denote by A( L, Γ) the space of affine functions.

• Theorem. (Potentials of locally exact forms) (CGIS 2010)

Any locally exact form ω ∈ Ω1

loc(K) admits a potential U ∈ A(

L, Γ) in the sense that

• γ

ω = U( γ(1)) − U( γ(0)). An affine function U ∈ A( L, Γ) is a potential of a locally exact form iff it has finite energy EΓ[U] := lim

n

5 3 n

e∈En

|U( e+) − U( e−)|2 .

Overview Sierpinski gasket Dirichlet form Differential calculus Fredholm modules, Conformal Geometry Dirac operators, Spectral triples Potential Theory

References

• A. Jonsson. A trace theorem for the Dirichlet form on the Sierpinski gasket,
• Math. Z., 250 (2005), 599–609.