Luke Rogers - Physics@Technion

Dirichlet forms and derivations on fractals
Luke Rogers
University of Connecticut
Luke Rogers Dirichlet forms and derivations on fractals

Goal: Quantize fractal analysis
There are ways to do analysis on certain fractal sets.
Typical examples include self-similar gaskets.
Heart of this analysis is Dirichlet form/Laplacian/Heat
semigroup.
Different properties than Euclidean: e.g. anomalous
diffusion, localized eigenfunctions.
Would like to quantize this analysis and obtain a quantum
theory for fractal substrate.
This talk is about an approach to doing this by following
some methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Goal: Quantize fractal analysis
There are ways to do analysis on certain fractal sets.
Typical examples include self-similar gaskets.
Heart of this analysis is Dirichlet form/Laplacian/Heat
semigroup.
Different properties than Euclidean: e.g. anomalous
diffusion, localized eigenfunctions.
Would like to quantize this analysis and obtain a quantum
theory for fractal substrate.
This talk is about an approach to doing this by following
some methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Goal: Quantize fractal analysis
There are ways to do analysis on certain fractal sets.
Typical examples include self-similar gaskets.
Heart of this analysis is Dirichlet form/Laplacian/Heat
semigroup.
Different properties than Euclidean: e.g. anomalous
diffusion, localized eigenfunctions.
Would like to quantize this analysis and obtain a quantum
theory for fractal substrate.
This talk is about an approach to doing this by following
some methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Goal: Quantize fractal analysis
There are ways to do analysis on certain fractal sets.
Typical examples include self-similar gaskets.
Heart of this analysis is Dirichlet form/Laplacian/Heat
semigroup.
Different properties than Euclidean: e.g. anomalous
diffusion, localized eigenfunctions.
Would like to quantize this analysis and obtain a quantum
theory for fractal substrate.
This talk is about an approach to doing this by following
some methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Goal: Quantize fractal analysis
There are ways to do analysis on certain fractal sets.
Typical examples include self-similar gaskets.
Heart of this analysis is Dirichlet form/Laplacian/Heat
semigroup.
Different properties than Euclidean: e.g. anomalous
diffusion, localized eigenfunctions.
Would like to quantize this analysis and obtain a quantum
theory for fractal substrate.
This talk is about an approach to doing this by following
some methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Analysis on fractals via scaling limits
We will get our Dirichlet form, Laplacian, and heat
semigroup as scaling limits from graphs.
Recall: Can get Brownian motion from rescaled Simple
Random Walk on integer lattice.
Similarly: Can get Laplacian as scaling limit of Laplacian
on lattice.
And classical Dirichlet form
R2 |∇u| 2 as scaling limit of
Dirichlet forms on lattice
Same idea works on some fractals, using graphs that
converge to the fractal.
For fractals, scaling is not the usual Euclidean scaling.
Luke Rogers Dirichlet forms and derivations on fractals

Analysis on fractals via scaling limits
We will get our Dirichlet form, Laplacian, and heat
semigroup as scaling limits from graphs.
Recall: Can get Brownian motion from rescaled Simple
Random Walk on integer lattice.
Similarly: Can get Laplacian as scaling limit of Laplacian
on lattice.
And classical Dirichlet form
R2 |∇u| 2 as scaling limit of
Dirichlet forms on lattice
Same idea works on some fractals, using graphs that
converge to the fractal.
For fractals, scaling is not the usual Euclidean scaling.
Luke Rogers Dirichlet forms and derivations on fractals

Analysis on fractals via scaling limits
We will get our Dirichlet form, Laplacian, and heat
semigroup as scaling limits from graphs.
Recall: Can get Brownian motion from rescaled Simple
Random Walk on integer lattice.
Similarly: Can get Laplacian as scaling limit of Laplacian
on lattice.
And classical Dirichlet form
R2 |∇u| 2 as scaling limit of
Dirichlet forms on lattice
Same idea works on some fractals, using graphs that
converge to the fractal.
For fractals, scaling is not the usual Euclidean scaling.
Luke Rogers Dirichlet forms and derivations on fractals

Analysis on fractals via scaling limits
We will get our Dirichlet form, Laplacian, and heat
semigroup as scaling limits from graphs.
Recall: Can get Brownian motion from rescaled Simple
Random Walk on integer lattice.
Similarly: Can get Laplacian as scaling limit of Laplacian
on lattice.
And classical Dirichlet form
R2 |∇u| 2 as scaling limit of
Dirichlet forms on lattice
Same idea works on some fractals, using graphs that
converge to the fractal.
For fractals, scaling is not the usual Euclidean scaling.
Luke Rogers Dirichlet forms and derivations on fractals

Analysis on fractals via scaling limits
We will get our Dirichlet form, Laplacian, and heat
semigroup as scaling limits from graphs.
Recall: Can get Brownian motion from rescaled Simple
Random Walk on integer lattice.
Similarly: Can get Laplacian as scaling limit of Laplacian
on lattice.
And classical Dirichlet form
R2 |∇u| 2 as scaling limit of
Dirichlet forms on lattice
Same idea works on some fractals, using graphs that
converge to the fractal.
For fractals, scaling is not the usual Euclidean scaling.
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Sierpinski Gasket
Em(u) =
2 u(y) − u(x)
y∼0x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Zero level graph
Dirichlet Form E0(u, u) =
2 u(y) − u(x)
y∼0x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
First level graph
Dirichlet Form E1(u, u) =
2 u(y) − u(x)
y∼1x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Second level graph
Dirichlet Form E2(u, u) =
2 u(y) − u(x)
y∼2x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Sierpinski Gasket
Dirichlet Form E(u, u) = lim
m→∞
5
m 2 u(y) − u(x)
3
y∼mx
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Zero level graph
Laplacian ∆0u(x) =
u(y) − u(x)
y∼0x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
First level graph
Laplacian ∆1u(x) =
u(y) − u(x)
y∼1x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Second level graph
Laplacian ∆2u(x) =
u(y) − u(x)
y∼2x
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Sierpinski Gasket
Laplacian ∆u(x) = 3
2 lim
m→∞
5m
y∼mx
u(y) − u(x)
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Sierpinski Gasket
Gauss-Green: E(u, v) = −
(∆u)v dµ
Luke Rogers Dirichlet forms and derivations on fractals

Scaling limits
Sierpinski Gasket
Gauss-Green: E(u, v) = −
(∆u)v dµ +
boundary
Luke Rogers Dirichlet forms and derivations on fractals
(du(x))v(x)

Harmonic and n-harmonic functions
If we prescribe values at 3 boundary vertices of Sierpinski
Gasket, then there is a unique function u with minimal
value of E(u, u) and those boundary values.
This minimizer is called harmonic. It has ∆u = 0.
If prescribe values at all scale n vertices, there is a unique
minimizer of E(u, u).
It is called n-harmonic and has ∆u = 0 except at scale n
vertices, where ∆u has point masses.
Space of n-harmonic functions is a good basis when
working with E as it is dense (when using E as inner
product). It is also dense in C(X) and L 2 (X).
Luke Rogers Dirichlet forms and derivations on fractals

Harmonic and n-harmonic functions
If we prescribe values at 3 boundary vertices of Sierpinski
Gasket, then there is a unique function u with minimal
value of E(u, u) and those boundary values.
This minimizer is called harmonic. It has ∆u = 0.
If prescribe values at all scale n vertices, there is a unique
minimizer of E(u, u).
It is called n-harmonic and has ∆u = 0 except at scale n
vertices, where ∆u has point masses.
Space of n-harmonic functions is a good basis when
working with E as it is dense (when using E as inner
product). It is also dense in C(X) and L 2 (X).
Luke Rogers Dirichlet forms and derivations on fractals

Harmonic and n-harmonic functions
If we prescribe values at 3 boundary vertices of Sierpinski
Gasket, then there is a unique function u with minimal
value of E(u, u) and those boundary values.
This minimizer is called harmonic. It has ∆u = 0.
If prescribe values at all scale n vertices, there is a unique
minimizer of E(u, u).
It is called n-harmonic and has ∆u = 0 except at scale n
vertices, where ∆u has point masses.
Space of n-harmonic functions is a good basis when
working with E as it is dense (when using E as inner
product). It is also dense in C(X) and L 2 (X).
Luke Rogers Dirichlet forms and derivations on fractals

Harmonic and n-harmonic functions
If we prescribe values at 3 boundary vertices of Sierpinski
Gasket, then there is a unique function u with minimal
value of E(u, u) and those boundary values.
This minimizer is called harmonic. It has ∆u = 0.
If prescribe values at all scale n vertices, there is a unique
minimizer of E(u, u).
It is called n-harmonic and has ∆u = 0 except at scale n
vertices, where ∆u has point masses.
Space of n-harmonic functions is a good basis when
working with E as it is dense (when using E as inner
product). It is also dense in C(X) and L 2 (X).
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Fractal Analysis
There is a good class of fractals for which approach on
preceding slide is applicable (post-critically finite self-similar).
When I say fractal analysis in this talk, I always have in mind
that situation:
A fractal set X.
A Dirichlet form E with domain F (all functions in domain
are continuous).
Equivalently a Laplacian ∆, or a heat flow e −t∆ , or a
Brownian motion.
Note: It is not clear how to apply this analysis to infinitely
ramified fractals like Sierpinski carpets.
Recall that (eventual) goal is to quantize fractal analysis using
methods from non-commutative geometry.
Luke Rogers Dirichlet forms and derivations on fractals

Non-commutative geometry: motivational sketch
Physical problem: In quantum world, co-ordinates do not
commute.
Solution: Replace co-ordinates by operators on Hilbert
space.
Problem: We also want fields.
Problem: Interesting spaces are curved.
Non-commutative geometry: Geometry and field theory on
“spaces” where co-ordinates (i.e. functions) do not
commute.
Luke Rogers Dirichlet forms and derivations on fractals

Non-commutative geometry: motivational sketch
Physical problem: In quantum world, co-ordinates do not
commute.
Solution: Replace co-ordinates by operators on Hilbert
space.
Problem: We also want fields.
Problem: Interesting spaces are curved.
Non-commutative geometry: Geometry and field theory on
“spaces” where co-ordinates (i.e. functions) do not
commute.
Luke Rogers Dirichlet forms and derivations on fractals

Non-commutative geometry: motivational sketch
Physical problem: In quantum world, co-ordinates do not
commute.
Solution: Replace co-ordinates by operators on Hilbert
space.
Problem: We also want fields.
Problem: Interesting spaces are curved.
Non-commutative geometry: Geometry and field theory on
“spaces” where co-ordinates (i.e. functions) do not
commute.
Luke Rogers Dirichlet forms and derivations on fractals

Non-commutative geometry: motivational sketch
Physical problem: In quantum world, co-ordinates do not
commute.
Solution: Replace co-ordinates by operators on Hilbert
space.
Problem: We also want fields.
Problem: Interesting spaces are curved.
Non-commutative geometry: Geometry and field theory on
“spaces” where co-ordinates (i.e. functions) do not
commute.
Luke Rogers Dirichlet forms and derivations on fractals

Non-commutative geometry: motivational sketch
Physical problem: In quantum world, co-ordinates do not
commute.
Solution: Replace co-ordinates by operators on Hilbert
space.
Problem: We also want fields.
Problem: Interesting spaces are curved.
Non-commutative geometry: Geometry and field theory on
“spaces” where co-ordinates (i.e. functions) do not
commute.
Luke Rogers Dirichlet forms and derivations on fractals

Geometry via algebra
Non-commutative geometry: write geometric notions (e.g.
curvature of manifold M) in terms of an algebra of
functions on M. (e.g. algebra C(M) of cts functions on M).
If the algebraic versions of these geometric ideas do not
require commutativity of the algebra one then has
“geometry” for non-commutative algebras of functions.
Goal: Do something similar for fractal analysis: express
analysis in terms of natural algebras of functions (e.g. cts
functions or functions of finite energy).
Next Step: replace the commutative algebras that come up
in analysis on fractals with physically meaningful
non-commutative ones.
Luke Rogers Dirichlet forms and derivations on fractals

Geometry via algebra
Non-commutative geometry: write geometric notions (e.g.
curvature of manifold M) in terms of an algebra of
functions on M. (e.g. algebra C(M) of cts functions on M).
If the algebraic versions of these geometric ideas do not
require commutativity of the algebra one then has
“geometry” for non-commutative algebras of functions.
Goal: Do something similar for fractal analysis: express
analysis in terms of natural algebras of functions (e.g. cts
functions or functions of finite energy).
Next Step: replace the commutative algebras that come up
in analysis on fractals with physically meaningful
non-commutative ones.
Luke Rogers Dirichlet forms and derivations on fractals

Geometry via algebra
Non-commutative geometry: write geometric notions (e.g.
curvature of manifold M) in terms of an algebra of
functions on M. (e.g. algebra C(M) of cts functions on M).
If the algebraic versions of these geometric ideas do not
require commutativity of the algebra one then has
“geometry” for non-commutative algebras of functions.
Goal: Do something similar for fractal analysis: express
analysis in terms of natural algebras of functions (e.g. cts
functions or functions of finite energy).
Next Step: replace the commutative algebras that come up
in analysis on fractals with physically meaningful
non-commutative ones.
Luke Rogers Dirichlet forms and derivations on fractals

Geometry via algebra
Non-commutative geometry: write geometric notions (e.g.
curvature of manifold M) in terms of an algebra of
functions on M. (e.g. algebra C(M) of cts functions on M).
If the algebraic versions of these geometric ideas do not
require commutativity of the algebra one then has
“geometry” for non-commutative algebras of functions.
Goal: Do something similar for fractal analysis: express
analysis in terms of natural algebras of functions (e.g. cts
functions or functions of finite energy).
Next Step: replace the commutative algebras that come up
in analysis on fractals with physically meaningful
non-commutative ones.
Luke Rogers Dirichlet forms and derivations on fractals

Geometry via algebra
Non-commutative geometry: write geometric notions (e.g.
curvature of manifold M) in terms of an algebra of
functions on M. (e.g. algebra C(M) of cts functions on M).
If the algebraic versions of these geometric ideas do not
require commutativity of the algebra one then has
“geometry” for non-commutative algebras of functions.
Goal: Do something similar for fractal analysis: express
analysis in terms of natural algebras of functions (e.g. cts
functions or functions of finite energy).
Next Step: replace the commutative algebras that come up
in analysis on fractals with physically meaningful
non-commutative ones.
Luke Rogers Dirichlet forms and derivations on fractals

Topology via algebra
Gelfand: the property of being a compact Hausdorff space can
be expressed in terms of algebra.
SPACE ALGEBRA
X compact Hausdorff A unital commutative C ∗
Isomorphisms:
X Hom(C(X), C) via x ↦→ point evaluation at x.
A C Hom(A, C) via a ↦→ â with â(φ) = φ(a) where a ∈ A and
φ ∈ Hom(A, C).
Luke Rogers Dirichlet forms and derivations on fractals

Topology via algebra
Gelfand: the property of being a compact Hausdorff space can
be expressed in terms of algebra.
SPACE ALGEBRA
X compact Hausdorff
A = C(X)
✲
✛
X = Hom(A, C)
A unital commutative C ∗
Isomorphisms:
X Hom(C(X), C) via x ↦→ point evaluation at x.
A C Hom(A, C) via a ↦→ â with â(φ) = φ(a) where a ∈ A and
φ ∈ Hom(A, C).
Luke Rogers Dirichlet forms and derivations on fractals

Topology via algebra
Gelfand: the property of being a compact Hausdorff space can
be expressed in terms of algebra.
SPACE ALGEBRA
X compact Hausdorff
A = C(X)
✲
✛
X = Hom(A, C)
A unital commutative C ∗
Isomorphisms:
X Hom(C(X), C) via x ↦→ point evaluation at x.
A C Hom(A, C) via a ↦→ â with â(φ) = φ(a) where a ∈ A and
φ ∈ Hom(A, C).
Luke Rogers Dirichlet forms and derivations on fractals

Riemannian geometry via algebra (Connes)
SPACE ALGEBRA
X compact spin manifold H Hilbert space
g Riemannian metric A algebra of operators on H
D self-adjoint operator on H
Some Properties
Forward map:
H is the spinor bundle
A = C(X)
D is Dirac operator (square root of Laplacian)
(H, A, D) called a “spectral triple’
Luke Rogers Dirichlet forms and derivations on fractals

Riemannian geometry via algebra (Connes)
SPACE ALGEBRA
X compact spin manifold
g Riemannian metric
Forward map:
H is the spinor bundle
✛
✲ H Hilbert space
A algebra of operators on H
D self-adjoint operator on H
Some Properties
A = C(X) (acting by multiplication)
D is Dirac operator (square root of Laplacian)
(H, A, D) called a “spectral triple”
Luke Rogers Dirichlet forms and derivations on fractals

Recovering distance from algebra
Define differential da = [D, a] = Da − aD.
1-Lipschitz function a has
op
[D, a] ≤ 1.
Distance sup |a(x) − a(y)| : all a ∈ A,
op
[D, a] ≤ 1 .
Luke Rogers Dirichlet forms and derivations on fractals

Recovering distance from algebra
Define differential da = [D, a] = Da − aD.
1-Lipschitz function a has
op
[D, a] ≤ 1.
Distance sup |a(x) − a(y)| : all a ∈ A,
op
[D, a] ≤ 1 .
Luke Rogers Dirichlet forms and derivations on fractals

Recovering distance from algebra
Define differential da = [D, a] = Da − aD.
1-Lipschitz function a has
op
[D, a] ≤ 1.
Distance sup |a(x) − a(y)| : all a ∈ A,
op
[D, a] ≤ 1 .
Luke Rogers Dirichlet forms and derivations on fractals

Recovering volume from algebra
Compact operators play role of infinitesimals
If T is a compact operator, λn(T ) is n th singular value
Integration is given by application of a trace operator, the
Dixmier trace, which is a version of
1
lim
N→∞ log N
N
λn(T )
0
Luke Rogers Dirichlet forms and derivations on fractals

Recovering volume from algebra
Compact operators play role of infinitesimals
If T is a compact operator, λn(T ) is n th singular value
Integration is given by application of a trace operator, the
Dixmier trace, which is a version of
1
lim
N→∞ log N
N
λn(T )
0
Luke Rogers Dirichlet forms and derivations on fractals

Recovering volume from algebra
Compact operators play role of infinitesimals
If T is a compact operator, λn(T ) is n th singular value
Integration is given by application of a trace operator, the
Dixmier trace, which is a version of
1
lim
N→∞ log N
N
λn(T )
0
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
Cipriani-Sauvageot (2007):
Start with fractal X having a Dirichlet form E.
Algebras of functions C(X) (continuous functions) and F
(those with E(a, a) < ∞).
Build Hilbert module H (1-forms).
Build derivation ∂ : F → H.
Has Leibniz rule ∂(ab) = a(∂b) + (∂a)b.
Point is that E(a, a) = ∂a 2
H .
(compare to Euclidean case E(a) = |∇a| 2 .)
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
Cipriani-Sauvageot (2007):
Start with fractal X having a Dirichlet form E.
Algebras of functions C(X) (continuous functions) and F
(those with E(a, a) < ∞).
Build Hilbert module H (1-forms).
Build derivation ∂ : F → H.
Has Leibniz rule ∂(ab) = a(∂b) + (∂a)b.
Point is that E(a, a) = ∂a 2
H .
(compare to Euclidean case E(a) = |∇a| 2 .)
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
Cipriani-Sauvageot (2007):
Start with fractal X having a Dirichlet form E.
Algebras of functions C(X) (continuous functions) and F
(those with E(a, a) < ∞).
Build Hilbert module H (1-forms).
Build derivation ∂ : F → H.
Has Leibniz rule ∂(ab) = a(∂b) + (∂a)b.
Point is that E(a, a) = ∂a 2
H .
(compare to Euclidean case E(a) = |∇a| 2 .)
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
Cipriani-Sauvageot (2007):
Start with fractal X having a Dirichlet form E.
Algebras of functions C(X) (continuous functions) and F
(those with E(a, a) < ∞).
Build Hilbert module H (1-forms).
Build derivation ∂ : F → H.
Has Leibniz rule ∂(ab) = a(∂b) + (∂a)b.
Point is that E(a, a) = ∂a 2
H .
(compare to Euclidean case E(a) = |∇a| 2 .)
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
Cipriani-Sauvageot (2007):
Start with fractal X having a Dirichlet form E.
Algebras of functions C(X) (continuous functions) and F
(those with E(a, a) < ∞).
Build Hilbert module H (1-forms).
Build derivation ∂ : F → H.
Has Leibniz rule ∂(ab) = a(∂b) + (∂a)b.
Point is that E(a, a) = ∂a 2
H .
(compare to Euclidean case E(a) = |∇a| 2 .)
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
SPACE ALGEBRA
X fractal
H Hilbert space
E Dirichlet form A algebra of operators on H
J phase operator on H
H is a quotient of F ⊗ F
A = C(X) (acting by multiplication)
J is sign of Dirac operator (so J = D|D| −1 )
(H, A, J) called a “spectral triple’
There is a derivation ∂ : F → H so E(a, a) = da 2
H
Luke Rogers Dirichlet forms and derivations on fractals

Spectral triple for fractal analysis
SPACE ALGEBRA
X fractal
E Dirichlet form
✛
✲ H Hilbert space
A algebra of operators on H
J phase operator on H
H is a certain quotient of F ⊗ F
A = C(X) (acting by multiplication)
J is phase of Dirac operator (so J = D|D| −1 )
(H, A, J) called a “spectral triple’
There is a derivation ∂ : F → H so E(a, a) = da 2
H
Luke Rogers Dirichlet forms and derivations on fractals

H and the derivation
The space H is supposed to be the 1-forms, but its definition is
somewhat abstract.
Construction:
Begin with space F ⊗ F and let
〈a ⊗ b, c ⊗ d〉H = 1
E(a, cdb) + E(abd, c) − E(bd, ac) .
2
Quotient out by zero-norm subspace. Take completion to
get H.
Define
a(b ⊗ c) = (ab) ⊗ c − a ⊗ (bc)
(b ⊗ c)d = b ⊗ (cd)
and extend to H by linearity, density and continuity.
Let ∂ : F → H be map a ↦→ a ⊗ 1.
Luke Rogers Dirichlet forms and derivations on fractals

H and the derivation
The space H is supposed to be the 1-forms, but its definition is
somewhat abstract.
Construction:
Begin with space F ⊗ F and let
〈a ⊗ b, c ⊗ d〉H = 1
E(a, cdb) + E(abd, c) − E(bd, ac) .
2
Quotient out by zero-norm subspace. Take completion to
get H.
Define
a(b ⊗ c) = (ab) ⊗ c − a ⊗ (bc)
(b ⊗ c)d = b ⊗ (cd)
and extend to H by linearity, density and continuity.
Let ∂ : F → H be map a ↦→ a ⊗ 1.
Luke Rogers Dirichlet forms and derivations on fractals

H and the derivation
The space H is supposed to be the 1-forms, but its definition is
somewhat abstract.
Construction:
Begin with space F ⊗ F and let
〈a ⊗ b, c ⊗ d〉H = 1
E(a, cdb) + E(abd, c) − E(bd, ac) .
2
Quotient out by zero-norm subspace. Take completion to
get H.
Define
a(b ⊗ c) = (ab) ⊗ c − a ⊗ (bc)
(b ⊗ c)d = b ⊗ (cd)
and extend to H by linearity, density and continuity.
Let ∂ : F → H be map a ↦→ a ⊗ 1.
Luke Rogers Dirichlet forms and derivations on fractals

H and the derivation
The space H is supposed to be the 1-forms, but its definition is
somewhat abstract.
Construction:
Begin with space F ⊗ F and let
〈a ⊗ b, c ⊗ d〉H = 1
E(a, cdb) + E(abd, c) − E(bd, ac) .
2
Quotient out by zero-norm subspace. Take completion to
get H.
Define
a(b ⊗ c) = (ab) ⊗ c − a ⊗ (bc)
(b ⊗ c)d = b ⊗ (cd)
and extend to H by linearity, density and continuity.
Let ∂ : F → H be map a ↦→ a ⊗ 1.
Luke Rogers Dirichlet forms and derivations on fractals

H and the derivation
The space H is supposed to be the 1-forms, but its definition is
somewhat abstract.
Construction:
Begin with space F ⊗ F and let
〈a ⊗ b, c ⊗ d〉H = 1
E(a, cdb) + E(abd, c) − E(bd, ac) .
2
Quotient out by zero-norm subspace. Take completion to
get H.
Define
a(b ⊗ c) = (ab) ⊗ c − a ⊗ (bc)
(b ⊗ c)d = b ⊗ (cd)
and extend to H by linearity, density and continuity.
Let ∂ : F → H be map a ↦→ a ⊗ 1.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding H with n-harmonic functions
Take a scale n cell Xα.
For prescribed values on boundary of this cell, take any
n-harmonic function hα with these values.
Then hα ⊗ 1α is an element of H. Think of as just hα on the
cell and identically zero elsewhere.
Scale n elements of H are sums of these functions; think
of them as harmonic on each of the scale n cells, but with
no matching conditions at the intersections of cells.
Any element of H is a limit (in H norm) of such scale n
functions.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding H with n-harmonic functions
Take a scale n cell Xα.
For prescribed values on boundary of this cell, take any
n-harmonic function hα with these values.
Then hα ⊗ 1α is an element of H. Think of as just hα on the
cell and identically zero elsewhere.
Scale n elements of H are sums of these functions; think
of them as harmonic on each of the scale n cells, but with
no matching conditions at the intersections of cells.
Any element of H is a limit (in H norm) of such scale n
functions.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding H with n-harmonic functions
Take a scale n cell Xα.
For prescribed values on boundary of this cell, take any
n-harmonic function hα with these values.
Then hα ⊗ 1α is an element of H. Think of as just hα on the
cell and identically zero elsewhere.
Scale n elements of H are sums of these functions; think
of them as harmonic on each of the scale n cells, but with
no matching conditions at the intersections of cells.
Any element of H is a limit (in H norm) of such scale n
functions.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding H with n-harmonic functions
Take a scale n cell Xα.
For prescribed values on boundary of this cell, take any
n-harmonic function hα with these values.
Then hα ⊗ 1α is an element of H. Think of as just hα on the
cell and identically zero elsewhere.
Scale n elements of H are sums of these functions; think
of them as harmonic on each of the scale n cells, but with
no matching conditions at the intersections of cells.
Any element of H is a limit (in H norm) of such scale n
functions.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding H with n-harmonic functions
Take a scale n cell Xα.
For prescribed values on boundary of this cell, take any
n-harmonic function hα with these values.
Then hα ⊗ 1α is an element of H. Think of as just hα on the
cell and identically zero elsewhere.
Scale n elements of H are sums of these functions; think
of them as harmonic on each of the scale n cells, but with
no matching conditions at the intersections of cells.
Any element of H is a limit (in H norm) of such scale n
functions.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding ∂ with n-harmonic functions
Any function a ∈ F (i.e. with E(a, a) < ∞) can be
approximated in sense of E by n-harmonic functions.
The map ∂ takes a to a ⊗ 1.
So if a is n-harmonic then ∂a = a ⊗ 1 and everything from
F can be approximated by these functions.
H is limits of functions harmonic on scale n cells, with no
matching conditions at vertices.
Image of ∂ is limits of functions harmonic on scale n cells,
with continuous matching at vertices.
Using the Gauss-Green formula (locally) we find that the
orthogonal space to image of ∂ is limits of functions
harmonic on scale n cells, with normal derivatives
summing to zero at vertices.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding ∂ with n-harmonic functions
Any function a ∈ F (i.e. with E(a, a) < ∞) can be
approximated in sense of E by n-harmonic functions.
The map ∂ takes a to a ⊗ 1.
So if a is n-harmonic then ∂a = a ⊗ 1 and everything from
F can be approximated by these functions.
H is limits of functions harmonic on scale n cells, with no
matching conditions at vertices.
Image of ∂ is limits of functions harmonic on scale n cells,
with continuous matching at vertices.
Using the Gauss-Green formula (locally) we find that the
orthogonal space to image of ∂ is limits of functions
harmonic on scale n cells, with normal derivatives
summing to zero at vertices.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding ∂ with n-harmonic functions
Any function a ∈ F (i.e. with E(a, a) < ∞) can be
approximated in sense of E by n-harmonic functions.
The map ∂ takes a to a ⊗ 1.
So if a is n-harmonic then ∂a = a ⊗ 1 and everything from
F can be approximated by these functions.
H is limits of functions harmonic on scale n cells, with no
matching conditions at vertices.
Image of ∂ is limits of functions harmonic on scale n cells,
with continuous matching at vertices.
Using the Gauss-Green formula (locally) we find that the
orthogonal space to image of ∂ is limits of functions
harmonic on scale n cells, with normal derivatives
summing to zero at vertices.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding ∂ with n-harmonic functions
Any function a ∈ F (i.e. with E(a, a) < ∞) can be
approximated in sense of E by n-harmonic functions.
The map ∂ takes a to a ⊗ 1.
So if a is n-harmonic then ∂a = a ⊗ 1 and everything from
F can be approximated by these functions.
H is limits of functions harmonic on scale n cells, with no
matching conditions at vertices.
Image of ∂ is limits of functions harmonic on scale n cells,
with continuous matching at vertices.
Using the Gauss-Green formula (locally) we find that the
orthogonal space to image of ∂ is limits of functions
harmonic on scale n cells, with normal derivatives
summing to zero at vertices.
Luke Rogers Dirichlet forms and derivations on fractals

Understanding ∂ with n-harmonic functions
Any function a ∈ F (i.e. with E(a, a) < ∞) can be
approximated in sense of E by n-harmonic functions.
The map ∂ takes a to a ⊗ 1.
So if a is n-harmonic then ∂a = a ⊗ 1 and everything from
F can be approximated by these functions.
H is limits of functions harmonic on scale n cells, with no
matching conditions at vertices.
Image of ∂ is limits of functions harmonic on scale n cells,
with continuous matching at vertices.
Using the Gauss-Green formula (locally) we find that the
orthogonal space to image of ∂ is limits of functions
harmonic on scale n cells, with normal derivatives
summing to zero at vertices.
Luke Rogers Dirichlet forms and derivations on fractals

DeRham cohomology and Hodge Theorem
The result on the previous page says that we can recover
the topology (the loops) of X from H.
In mathematical terms, the map ∂ is the exterior derivative
for a DeRham complex, and a version of the Hodge
theorem is true.
One consequence: Map ∂ is surjective iff there are no
loops, i.e. X is a tree.
Luke Rogers Dirichlet forms and derivations on fractals

DeRham cohomology and Hodge Theorem
The result on the previous page says that we can recover
the topology (the loops) of X from H.
In mathematical terms, the map ∂ is the exterior derivative
for a DeRham complex, and a version of the Hodge
theorem is true.
One consequence: Map ∂ is surjective iff there are no
loops, i.e. X is a tree.
Luke Rogers Dirichlet forms and derivations on fractals

DeRham cohomology and Hodge Theorem
The result on the previous page says that we can recover
the topology (the loops) of X from H.
In mathematical terms, the map ∂ is the exterior derivative
for a DeRham complex, and a version of the Hodge
theorem is true.
One consequence: Map ∂ is surjective iff there are no
loops, i.e. X is a tree.
Luke Rogers Dirichlet forms and derivations on fractals

The phase map J
Map J is supposed to be the phase of the Dirac operator.
Let P be projection in H onto image of ∂ and P ⊥ the
orthogonal projection.
Define J = P − P ⊥ .
We can understand J very explicitly using preceding
description of ∂.
Luke Rogers Dirichlet forms and derivations on fractals

The phase map J
Map J is supposed to be the phase of the Dirac operator.
Let P be projection in H onto image of ∂ and P ⊥ the
orthogonal projection.
Define J = P − P ⊥ .
We can understand J very explicitly using preceding
description of ∂.
Luke Rogers Dirichlet forms and derivations on fractals

The phase map J
Map J is supposed to be the phase of the Dirac operator.
Let P be projection in H onto image of ∂ and P ⊥ the
orthogonal projection.
Define J = P − P ⊥ .
We can understand J very explicitly using preceding
description of ∂.
Luke Rogers Dirichlet forms and derivations on fractals

Commutator [J, a]
Initially consider commutator [J, a] of J with multiplication
by a ∈ C(X).
We can permit discontinuities of a at vertices of cells.
If a is constant on scale n cells then [J, a]h = 0 for any h
that is orthogonal to n-harmonic functions
Can approximate any [J, a] in operator norm by operators
with finite dimensional image
Hence [J, a] is a compact operator.
Luke Rogers Dirichlet forms and derivations on fractals

Commutator [J, a]
Initially consider commutator [J, a] of J with multiplication
by a ∈ C(X).
We can permit discontinuities of a at vertices of cells.
If a is constant on scale n cells then [J, a]h = 0 for any h
that is orthogonal to n-harmonic functions
Can approximate any [J, a] in operator norm by operators
with finite dimensional image
Hence [J, a] is a compact operator.
Luke Rogers Dirichlet forms and derivations on fractals

Commutator [J, a]
Initially consider commutator [J, a] of J with multiplication
by a ∈ C(X).
We can permit discontinuities of a at vertices of cells.
If a is constant on scale n cells then [J, a]h = 0 for any h
that is orthogonal to n-harmonic functions
Can approximate any [J, a] in operator norm by operators
with finite dimensional image
Hence [J, a] is a compact operator.
Luke Rogers Dirichlet forms and derivations on fractals

Commutator [J, a]
Initially consider commutator [J, a] of J with multiplication
by a ∈ C(X).
We can permit discontinuities of a at vertices of cells.
If a is constant on scale n cells then [J, a]h = 0 for any h
that is orthogonal to n-harmonic functions
Can approximate any [J, a] in operator norm by operators
with finite dimensional image
Hence [J, a] is a compact operator.
Luke Rogers Dirichlet forms and derivations on fractals

Commutator [J, a]
Initially consider commutator [J, a] of J with multiplication
by a ∈ C(X).
We can permit discontinuities of a at vertices of cells.
If a is constant on scale n cells then [J, a]h = 0 for any h
that is orthogonal to n-harmonic functions
Can approximate any [J, a] in operator norm by operators
with finite dimensional image
Hence [J, a] is a compact operator.
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability of [J, a]
Compact operator T = [J, a] is an “infinitesimal”.
Let λn(T ) be the singular values (note λn → 0)
Study zeta function n λ s n .
Can control λn using oscillation of a on m-cells, where
n ≍ e mdH (Hausdorff dimension).
Crude bounds show zeta function converges for ℜ(s) > d S
(spectral dimension).
This region of convergence was earlier proved by
Cipriani-Sauvageot via heat kernel methods
Luke Rogers Dirichlet forms and derivations on fractals

Summability below Spectral dimension
For a more restricted class of fractals the zeta function for
[J, a] converges below the spectral dimension if a is
n-harmonic.
Naotaka Kajino has pointed out that in the case of the
Sierpinski Gasket the spectral dimension for the gasket in
harmonic coordinates may be more natural than the
spectral dimension in the usual coordinates; we have not
yet checked this.
Our proof is suggestive of the possibility that the “correct”
condition for convergence of the zeta function at s may be
that a is in a Besov space with exponent depending on s.
Perhaps an analogue of a result of Connes-Sullivan for
quasicircles.
Luke Rogers Dirichlet forms and derivations on fractals

Summability below Spectral dimension
For a more restricted class of fractals the zeta function for
[J, a] converges below the spectral dimension if a is
n-harmonic.
Naotaka Kajino has pointed out that in the case of the
Sierpinski Gasket the spectral dimension for the gasket in
harmonic coordinates may be more natural than the
spectral dimension in the usual coordinates; we have not
yet checked this.
Our proof is suggestive of the possibility that the “correct”
condition for convergence of the zeta function at s may be
that a is in a Besov space with exponent depending on s.
Perhaps an analogue of a result of Connes-Sullivan for
quasicircles.
Luke Rogers Dirichlet forms and derivations on fractals

Summability below Spectral dimension
For a more restricted class of fractals the zeta function for
[J, a] converges below the spectral dimension if a is
n-harmonic.
Naotaka Kajino has pointed out that in the case of the
Sierpinski Gasket the spectral dimension for the gasket in
harmonic coordinates may be more natural than the
spectral dimension in the usual coordinates; we have not
yet checked this.
Our proof is suggestive of the possibility that the “correct”
condition for convergence of the zeta function at s may be
that a is in a Besov space with exponent depending on s.
Perhaps an analogue of a result of Connes-Sullivan for
quasicircles.
Luke Rogers Dirichlet forms and derivations on fractals