Waves, energy on fractals and related questions

A.Teplyaev

University of Connecticut

5th Cornell Conference on Analysis, Probability, andMathematical Physics on Fractals

June 11–15, 2014

Nuclear Physics B280 [FS 18] (1987) 147-180 North-Holland, Amsterdam

METRIC SPACE-TIME AS FIXED POINT

OF THE RENORMALIZATION GROUP EQUATIONS

ON FRACTAL STRUCTURES

F. ENGLERT, J.-M. FRI~RE x and M. ROOMAN 2

Physique Thkorique, C.P. 225, Universitb Libre de Bruxelles, 1050 Brussels, Belgium

Ph. SPINDEL

Facultb des Sciences, Universitb de l'Etat it Mons, 7000 Mons, Belgium

Received 19 February 1986

We take a model of foamy space-time structure described by self-similar fractals. We study the propagation of a scalar field on such a background and we show that for almost any initial conditions the renormalization group equations lead to an effective highly symmetric metric at large scale.

1. Introduction

Quantum gravity presents a potential difficulty which persists in any unification

program which incorporates gravity in the framework of a local field theory in

dimensions d > 4. In all such theories a local O ( d - 1 , 1 ) space-time symmetry is quite generally assumed at the outset as a "kinematical" symmetry of the classical

action. Such an extrapolation from relatively large distances, where the symmetry

0(3 ,1) is tested to a genuine local property is questionable. Indeed, the unbounded- ness of the Einstein curvature term in the analytically continued euclidean action

signals violent fluctuations near the Planck scale. Hence a "foamy" fractal space-time structure is expected [1], from which the average metric below this scale should emerge in a dynamical way. There is no obvious reason why a smooth effective

metric should at all be generated, and even if it were, why it should bear any relation to the "bare" symmetrical local metric imposed on the "fundamental"

1 Chercheur qualifi~ du FNRS. 2 Chercheur IISN.

0619-6823/87/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

6/14/2014 François Englert - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Fran%C3%A7ois_Englert 1/5

François Englert

François Englert in Israel, 2007

Born 6 November 1932

Etterbeek, Brussels, Belgium[1]

Nationality Belgian

Fields Theoretical physics

Institutions Université Libre de Bruxelles

Tel Aviv University[2][3]

Alma mater Université Libre de Bruxelles

Notable awards Francqui Prize (1982)

Wolf Prize in Physics (2004)

Sakurai Prize (2010)

Nobel Prize in Physics (2013)

François EnglertFrom Wikipedia, the free encyclopedia

François Baron Englert (French: [ɑɡlɛʁ]; born 6 November1932) is a Belgian theoretical physicist and 2013 Nobel prizelaureate (shared with Peter Higgs). He is Professor emeritusat the Université libre de Bruxelles (ULB) where he ismember of the Service de Physique Théorique. He is also aSackler Professor by Special Appointment in the School ofPhysics and Astronomy at Tel Aviv University and a memberof the Institute for Quantum Studies at Chapman University inCalifornia. He was awarded the 2010 J. J. Sakurai Prize forTheoretical Particle Physics (with Gerry Guralnik,C. R. Hagen, Tom Kibble, Peter Higgs, and Robert Brout),the Wolf Prize in Physics in 2004 (with Brout and Higgs) andthe High Energy and Particle Prize of the European PhysicalSociety (with Brout and Higgs) in 1997 for the mechanismwhich unifies short and long range interactions by generatingmassive gauge vector bosons. He has made contributions instatistical physics, quantum field theory, cosmology, string

theory and supergravity.[4] He is the recipient of the 2013Prince of Asturias Award in technical and scientific research,together with Peter Higgs and the CERN.

Englert was awarded the 2013 Nobel Prize in Physics,together with Peter Higgs for the discovery of the Higgs

mechanism.[5]

Contents

1 Early life

2 Academic career

3 The Brout–Englert–Higgs–Guralnik–Hagen–

Kibble mechanism[7]

4 Major awards

5 References

6 External links

Early life

150 F. Englert et al. / Metric space-time

Fig. 1. The first two iterations of a 2-dimensional 3-fractal.

tive integers v i (i = 1 . . . . . d) such that their s u m Y~./d=lP i is less or equal to n. All these points are contained in the hypertetrahedron bounded by the coordinate hyperplanes and the E~a=lVi = n hyperplane. We distinguish interior points and points belonging to a k-face (k < d), that is points characterized by a set of coordinates vj which contains d - k subsets s such that ~ , ~svi = 0 (mod n). Every point belongs to the boundary of at least one sub-hypertetrahedron and two points are called neighbours if they belong to the same sub-hypertetrahedron. One goes from a point to one of its neighbours by one of the elementary translations t i and lij defined as:

_+ ti: v~--+ v~: , where v~ = v k if k :~ i,

v" = v i + 1 ;

l q : v--+ vj , w h e r e v'k = v k i f i 4= k --t= j ;

v" = v i + 1,

v~ = v j - 1. (2.1)

In general, an interior point admits d ( d + 1) neighbours reached by the 2d transla- tions ___t i and the d ( d - 1 ) l q translations. If a point belongs to a k-face of the hypertetrahedron, some of these operations reach a point outside the initial hyperte- trahedron. Actually, points belonging to a k-face have only d ( k + 1) neighbours.

MIN KOWSKIAN

METRICS

EUCLIDEAN METRICS

/ /

/

/ /

/ /

/ /

/ /

/ /

/

\ \

N N

i~ °

fix

i I ' x I \ I " x

EUCLIDEAN

11

METRICS 03 ~ '~

c ~ = - P . . ~-

/3+1

Fig. 5. The plane of 2-parameter homogeneous metrics on the Sierpinski gasket. The hyperbole a = /3/(,8 + 1) separates the domain of euclidean metrics from minkowskian metrics and corresponds - except at the origin - to 1-dimensional metrics. ML, M 2, Ma denote unstable minkowskian fixed geometries while E corresponds to the stable euclidean fixed point. The unstable fixed points 01, 0 2 and 0 3 associated to 0-dimensional geometries are located at the origin and at infinity on the (a, /~) coordinates axis. The six straight lines are subsets invariant with respect to the recursion relation but repulsive in the region where they are dashed. The first points of two sequences of iterations are drawn. Note that for one of

them the 10th point (a = -56 .4 , /3 = -52 .5) is outside the frame of the figure.

F. Englert et al. / Metric space-time

I , I

177

1,1 , I I

0 II,II

Fig. 10. A metrical representation of the two first iterations of a 2-dimensional 2-fractal corresponding to the euclidean fixed point. Vertices are labelled according to fig. 4.

angles of the cell without its base, that is 57r, minus the sum of the angles not belonging to the cell and touching the 3 exterior vertices of the cell, that is 6~r - ~r = 5~r. We find thus that the curvature of a cell is zero, which is consistent with the assumption that the space surrounding the cell is flat.

Though the exact value of the curvature at each vertex of a cell is subject to some arbitrariness, because of the arbitrariness showed in the previous section of the normalization of the ?~i9's at successive levels, one easily verifies that, for the homogeneous metrics considered here, all the non-zero cancelling curvatures are located at the cell boundaries. The vertices belonging to the p and (p + 1) levels ot fractalization have negative curvature, the others have positive curvature.

Consider now a metric n-fractal, n >> 1, cutoff after the first iteration (or equivalently a ( p - 1) triangle in a fractal cutoff at the p th level). The result is a triangular lattice. Because the integrated curvature of any cell is zero, the inside of the lattice is correctly described on the average by a locally flat metric. From

170 R. MEYERS, R.S. STRICHARTZ AND A. TEPLYAEV

DDDDDDD

DDDD

``````````

\\\\\\\\\\\

rrr

rrr r

q1 q3

q2aa1

a2a3

xy

zFigure 6.4. Geometric interpretation of Proposition 6.1.

7. Effective resistance metric, Green’s functionand capacity of points

We first recall from [Ki4] some facts about limits of resistance networks.Although we state all the results of this section for the Sierpinski gasket,they can be applied to general pcf fractals with only minor changes.

Let E(f, f) be defined by (1.2) for any function f on V∗, where En is acompatible sequence of Dirichlet forms on Γn.

Proposition 7.1. Every point of V∗ =⋃

n≥0 Vn has positive capacity.

Proof. Let x ∈ V∗. Then x ∈ Vn for some n. The capacity of x with respectto E is the same as that with respect to En because of the compatibility ofthe sequence of networks. The latter capacity is positive because Vn is afinite set.

The effective resistance is defined for any x, y ∈ V∗ by

R(x, y) =(minuE(u, u) : u(x) = 1, u(y) = 0

)−1.(7.1)

Here the minimum is taken over all functions on V∗. Note that x, y ∈ Vnfor large enough n and that (7.1) does not change if E is replaced by En,because of the compatibility condition (see [Ki4], Proposition 2.1.11). ByTheorem 2.1.14 in [Ki4], R(x, y) is a metric on V∗. In what follows we willwrite R-continuity, R-closure etc. for continuity, closure etc. with respect tothe effective resistance metric R. It is known that if E(u, u) < ∞ then u isR-continuous ([Ki4], Theorem 2.2.6(1)). The main ingredient in the proofof this fact is the inequality

|u(x)− u(y)|2 ≤ R(x, y)E(u, u).(7.2)

Let Ω be the R-completion of V∗. We can conclude from (7.2) that if uis a function on V∗ such that E(u, u) <∞ then u has a unique continuation

Table of Contents

Remarks on

Local derivatives on fractals (theoretical)

Liouville Quantum Gravity as a fractal (experimental)

Waves on one dimensional fractals (theoretical/experimental)

Waves propagate faster with increased fractality

Ref talks ofMichael HinzDan KelleherLuke Rogersconcerning Dirac and magnetic Laplacian operators,and Sobolev spaces on fractals. See also

Rogers, Estimates for the resolvent kernel of the Laplacian onp.c.f. self-similar fractals and blowups. TAMS (2012)

Ionescu, Pearse, Rogers, Ruan, Strichartz, The resolvent kernelfor PCF self-similar fractals. TAMS (2010)

DeGrado, Rogers, Strichartz, Gradients of Laplacianeigenfunctions on the Sierpinski gasket. PAMS (2009)

Differential forms on fractals,Vector equations

div(a(∇u)) = f (1)

∆u + b(∇u) = f (2)

i∂u

∂t= (−i∇− A)2u + Vu. (3)

∂u∂t + (u · ∇)u −∆u +∇p = 0,

div u = 0,(4)

Hodge and Navier-Stokes on fractal

Theorem (a Hodge theorem, Hinz-T)

If the space is compact, connected and topologicallyone-dimensional of arbitrarily large Hausdorff and spectraldimensions, then any 1-form ω ∈ H is harmonic if and only if it isin (Im ∂)⊥, that is divω = 0.

Using (u · ∇)u = 12∇|u|2 − u × curl u = 1

2∂ΓH(u) + 0, equation (4)becomes

div u = ∂∗u = 0

∆1u = 0, 12∂ΓH(u) + ∂p = 0, ∂u

∂t = 0(5)

TheoremAny weak solution u of (5) is unique, harmonic and stationary (i.e.ut = u0 is independent of t ∈ [0,∞)) for any divergence free initialcondition u0.

Liouville Quantum Gravity as a fractalresearch in progress (numerical) withGrigory Bonik and Joe Chen

E(f , f ) =

∫

[0,1]2|∇f |2dx =

∫

[0,1]2f ∆µfdµ

where µ is a random measure defined (as a weak limit) by

µ = lim const · exp γX (x) dx

where X (x) is the Gaussian free field.

Liouville heat kernel: regularity and bounds

Maillard P.∗, Rhodes R.†, Vargas V.‡, Zeitouni O.§

June 1, 2014

Abstract

We initiate in this paper the study of analytic properties of the Liouville heat kernel.In particular, we establish regularity estimates on the heat kernel and derive non triviallower and upper bounds.

Key words or phrases: Liouville quantum gravity, heat kernel, Liouville Brownian motion, Gaussian multi-plicative chaos.

MSC 2000 subject classifications: 35K08, 60J60, 60K37, 60J55, 60J70

Contents

1 Introduction 2

2 Setup 62.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Log-correlated Gaussian fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Liouville measure and Liouville Brownian motion . . . . . . . . . . . . . . . . . 6

3 Representation and regularity of the heat kernel on the torus 9

4 Upper bounds on the heat kernel 134.1 Reminder on heat kernel estimates . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 The upper bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 Proof of Theorem 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

∗Weizmann Institute of Science, Rehovot, Israel. Partially supported by a grant from the Israel ScienceFoundation†Universite Paris-Dauphine, Ceremade, F-75016 Paris, France. Partially supported by grant ANR-11-JCJC

CHAMU‡Ecole Normale Superieure, DMA, 45 rue d’Ulm, 75005 Paris, France. Partially supported by grant ANR-

11-JCJC CHAMU§Weizmann Institute of Science, Rehovot, Israel. Partially supported by a grant from the Israel Science

Foundation

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and then obtain estimates on the heat kernel; this can be seen as a first step in a more ambitiousprogram devoted to the derivation of precise estimates on pγt (x, y). Note that the continuity weprove, the fact that the support of Mγ is full and the strict positivity of pγt (x, y) allows one todefine the Liouville Brownian bridge between any fixed x and y.

We recall that, on a standard smooth Riemannian manifold, Gaussian heat kernels estimatesin terms of the associated Riemannian distance have been established, see [22] for a review.Thereafter, heat kernel estimates have been obtained in the more exotic context of diffusions on(scale invariant) fractals: see [4, 25] for instance. In the context of LQG, it is natural to wonderwhat is the shape of the heat kernel and it is difficult to draw a clear expected picture: firstbecause of the multifractality of the geometry and second because the existence of the distancedγ associated to (1.1) remains one of the main open questions in LQG (though there has been

some progress in the case of pure gravity, i.e. γ =√

83: see [32] where the authors construct the

analog of growing quantum balls without proving the existence of the distance dγ).

Brief description of the results

Our main lower bound on the heat kernel reads as follows: if x, y and η > 0 are fixed, one canfind a random time T0 > 0 (depending on the GFF X and x, y, η) such that, for any t ∈]0, T0],

pγt (x, y) > exp(− t−

11+γ2/4−η

).

This is the content of Theorem 5.2 below. We emphasize that the exponent 1/(1 + γ2/4) is notexpected to be optimal, as in our derivation we do not take into account the geometry of theGaussian field.1

For γ2 6 4/3, the same heat kernel lower bound holds when the endpoints are sampledaccording to the measure Mγ, see subsection 5.3. The boundary γ2 6 4/3 is artificial and canbe improved to γ2 6 8/3, and with a worse lower bound, for γ2 > 8/3. We omit the details inthis version of the article.

We will also give the following uniform upper bound on the heat kernel (see Theorem 4.2below for a precise statement): for all δ > 0 there exists β = βδ(γ) and some random constantsc1, c2 > 0 (depending on the GFF X only) such that

∀x, y ∈ T, t > 0, pγt (x, y) 6( c1

t1+δ+ 1)

exp(− c2

(dT(x, y)

t1/β

) ββ−1 )

.

where dT is the standard distance on the torus. There is a gap between our lower and upperbound for the (inverse) power coefficient of t in the exponential : see figure 1 for a plot of2 + γ2/4 and β as a function of γ (the β in the plot corresponds to the limit of βδ as δ goes to0).

Note that our estimates on the heat kernel, and in particular the upper bound, are givenin terms of the Euclidean distance. The lower/upper bounds that we obtain do not match but

1Jian Ding and Marek Biskup have kindly shown us an argument based on ongoing works that allows one totake into account the geometry, and potentially may improve the exponent. The details will appear elsewhere.

3

this does not come as a surprise because such a matching would mean in a way that dγ (dT)θ

for some exponent θ > 0, which is not expected. Yet our results illustrate that we can read offthe Liouville heat kernel some uniform Holder control of the geometry of LQG in terms of theEuclidean geometry. This was already known for the Liouville measure by means of multifractalanalysis (see [35] for a precise statement and further references). To our knowledge, our work isone of the first to investigate the problem of heat kernel estimates in a multifractal context, insharp contrast with the monofractal framework of diffusions on fractals. Notice however thaton-diagonal heat kernel estimates have also been investigated in the context of one-dimensionalmultifractal geometry, see [3].

We conclude with some cautionary remarks on the (non)-sharpness of our methods. In boththe lower and upper bound, we have not taken much advantage of the geometry determinedby the GFF. In particular, our upper bounds are uniform on the torus, and thus certainly nottight for typical points. Similarly, in the derivation of our lower bound, we essentially force theLBM to follow a straight line between the starting and ending points. It is natural to expectthat forcing the LBM to follow a path adapted to the geometry of the GFF could yield a betterlower bound.

Discussion and speculations

Here we develop a short speculative discussion that has motivated at least partly our study. Itis concerned with the form of pγt (x, y) as t goes to 0 with x, y fixed, which arguably should berelated to the Hausdorff dimension dH(γ) of the metric space (T,dγ).

By analogy with the literature on fractals, it is natural to conjecture2 the following asymp-totic expression (t→ 0)

pγt (x, y) C

tdH (γ)

β

exp(− cdγ(x, y)

ββ−1

t1

β−1

). (1.2)

where C, c > 0 are some global constants (possibly random), β > 0 some exponent and meansthat pγ is bounded from above and below by two such expressions with possibly different valuesof c, C. Relation (1.2) should be understood for t less than some random threshold T (dependingon the free field X) as it could be the case that, for different regimes of t, the heat kernel is

controlled by the measure Mγ rather that the conjectural distance dγ. The ratio 2dH(γ)β

, calledthe spectral dimension of LQG, is equal to 2: this has been heuristically computed by Ambjørnand al. in [1] and then rigorously derived in a weaker form in [37]. This yields the relation

dH(γ) = β.

A general formula for dH(γ) has been proposed in the literature by Watabiki [41]

dH(γ) = 1 +γ2

4+

√(1 +

γ2

4

)2+ γ2. (1.3)

2Our results do not shed light on this conjecture, nor on whether the various parameters in (1.2) might bedifferent for points x, y sampled according to Mγ .

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3Spectral dimension of Liouville quantum gravity

Remi Rhodes ∗ Vincent Vargas †

Abstract

This paper is concerned with computing the spectral dimension of 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case ofboundary Liouville quantum gravity. We prove that the spectral dimensionis 1 via an exact expression for the boundary Liouville Brownian motion andheat kernel. Then we treat the 2d-case via a decomposition of time integraltransforms of the Liouville heat kernel into Gaussian multiplicative chaos ofBrownian bridges. We show that the spectral dimension is 2 in this case, asannounced by physicists (see Ambjørn and al. in [1]) fifteen years ago.

Key words or phrases: Liouville quantum gravity, Liouville Brownian motion, Gaussian multi-plicative chaos, Liouville heat kernel, spectral dimension.

Contents

1 Introduction 2

2 Boundary Liouville quantum gravity 32.1 Reminder about 1d-Riemannian structures . . . . . . . . . . . . . . . 42.2 Standard phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Critical phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 2d-Liouville quantum gravity 63.1 Brownian bridge decomposition . . . . . . . . . . . . . . . . . . . . . 73.2 Spectral dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

∗Universite Paris-Dauphine, Ceremade, F-75016 Paris, France. Partially supported by grantANR-11-JCJC CHAMU

†Universite Paris-Dauphine, Ceremade, F-75016 Paris, France. Partially supported by grantANR-11-JCJC CHAMU

1

Conjectures/Conclusions

Our numerical experiments on the trace of the heat kernel and theeigenvalue counting functions suggest

ds ∼ 1 if γ ∼ 2− ε

which implies that

1. the discrete approximations converge

2. the limit may not be unique

3. in view of the recent results by Kigami the measure µ mayhave no Volume Doubling property

Math. Proc. Camb. Phil. Soc. (2002), 132, 555 c© 2002 Cambridge Philosophical Society

DOI: 10.1017/S0305004101005618 Printed in the United Kingdom

555

Multifractal formalisms for the local spectraland walk dimensions

By B. M. HAMBLY†Mathematical Institute, University of Oxford, 24–29 St Giles, Oxford, OX1 3LB.

e-mail: hambly@maths.ox.ac.uk

JUN KIGAMI

Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan.e-mail: kigami@i.kyoto-u.ac.jp

and TAKASHI KUMAGAI‡Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.

e-mail: kumagai@kurims.kyoto-u.ac.jp

(Received 21 July 2000; revised 1 November 2000)

Abstract

We introduce the concepts of local spectral and walk dimension for fractals. For aclass of finitely ramified fractals we show that, if the Laplace operator on the fractal isdefined with respect to a multifractal measure, then both the local spectral and walkdimensions will have associated non-trivial multifractal spectra. The multifractalspectra for both dimensions can be calculated and are shown to be transformationsof the original underlying multifractal spectrum for the measure, but with respectto the effective resistance metric.

1 Introduction

Multifractal analysis was introduced in the physics literature (see for example[10]) to provide a finer description of fractal phenomena which displayed a rangeof power law scalings. A mathematically rigorous version has been developed in anumber of papers (see for example [1, 5, 6, 17, 18]). We give a brief discussion. Thelocal dimension of a measure µ at a point x in some fractal set K is defined to be, ifthe limit exists,

dimloc(x) = limr→0

logµ(Br(x))log r

,

where Br(x) is a ball of radius r about x. The sets of interest in multifractal analy-sis are Eα = x ∈ K: dimloc(x) exists and equals α. The multifractal spectrum isthen defined to be the function f (α) = dim(Eα), for some notion of dimension. A

† Partially supported by a Royal Society Industry Fellowship at BRIMS, HP-Labs.‡ Partially supported by Grant-in-Aid for Scientific Research (B)(2) 10440029.

E l e c t r o n ic

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Vol. 6 (2001) Paper no. 9, pages 1–23

Journal URLhttp://www.math.washington.edu/~ejpecp/

Paper URLhttp://www.math.washington.edu/~ejpecp/EjpVol6/paper9.abs.html

TRANSITION DENSITY ASYMPTOTICSFOR SOME DIFFUSION PROCESSES

WITH MULTI-FRACTAL STRUCTURES

Martin T. BarlowDepartment of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada

barlow@math.ubc.ca

Takashi KumagaiResearch Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

kumagai@kurims.kyoto-u.ac.jp

Abstract We study the asymptotics as t → 0 of the transition density of a class of µ-symmetricdiffusions in the case when the measure µ has a multi-fractal structure. These diffusions includesingular time changes of Brownian motion on the unit cube.

Keywords diffusion process, heat equation, transition density, spectral dimension, multi-fractal

AMS subject classification 60J60, 31C25, 60J65.

Research partially supported by a NSERC (Canada) grant and Grant-in-Aid for Scientific Re-search (B)(2) 10440029 of Japan.

Submitted to EJP on June 19, 2000. Final version accepted on March 3, 2001.

Waves on one dimensional fractals(theoretical/experimental)

J.F.-C.Chan, S.-M.Ngai, A.T., One-dimensional wave equationsdefined by fractal Laplacians. Journal d’Analyse Mathematique,to appear.

U.Andrews, J.P.Chen, G.Bonik, R.W.Martin, A.T., Waveequation on one-dimensional fractals with spectraldecimation. preprint

• Strichartz, A priori estimates for the wave equation andsome applications. J. Functional Analysis 5 (1970)....• Dalrymple, Strichartz, Vinson, Fractal differential equationson the Sierpinski gasket. J. Fourier Anal. Appl. 5 (1999)• Gibbons, Raj, Strichartz, The finite element method on theSierpinski gasket. Constr. Approx. 17 (2001)• Coletta, Dias, Strichartz, Numerical analysis on the Sierpinskigasket, with applications to Schrodinger equations, waveequation, and Gibbs’ phenomenon. Fractals 12 (2004)• Strichartz, Waves are recurrent on noncompact fractals. J.Fourier Anal. Appl. 16 (2010)• Constantin, Strichartz, Wheeler, Analysis of the Laplacian andspectral operators on the Vicsek set. Commun. Pure Appl.Anal. 10 (2011)

Consider same Laplacian as in the talk of Uta Freiberg, andhyperbolic initial/boundary value problem (IBVP):

utt −∆µu = f on [a, b]× [0,T ],

u = 0 on a, b × [0,T ],

u = g , ut = h on [a, b]× t = 0.(6)

If g ∈ Dom E , h ∈ L2µ[a, b] and f ∈ L2(0,T ; Dom E). Then

equation (6) has a unique weak solution.

We can consider weak forms, such as

−∫ b

aux(x , t) v ′(x) dx =

∫ b

autt(x , t) v(x) dµ, (7)

where ux(x , t) is the weak partial derivative of u with respect to xand utt is the weak second partial derivative with respect to t.Let µ be a self-similar measure defined by a one-dimensional IFSthat satisfies a family of second-order self-similar identities. Thenthe finite element method for the equation (7) discretizes it to asystem of second-order ordinary differential equations, which has aunique solution and can be solved numerically.Theorem. Let f = 0 in equation (6). Then the approximatesolutions um obtained by the finite element method converge inL2µ[a, b] to the actual weak solution u and

‖um − u‖µ ≤(C√

T ‖utt‖2,Dom E + 2 ‖u‖Dom E)ρm/2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

The weighted Bernoulli-type measure associated with the weightsp = 2−

√3 and 1− p =

√3− 1. The initial data g = sin(πx) and

h = 0 are used, Animations for this and other graphs in the paperare created and uploaded to the webpagehttp://homepages.uconn.edu/fractals/wave/.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

The infinite Bernoulli convolution associated with the golden ratio

is defined by the IFS S1(x) = ρx , S2(x) = ρx + (1− ρ), ρ =√5−12 .

0 1

S1 AAUS2

1− ρ ρ0 1

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

The 3-fold convolution of the Cantor measure satisfies a family ofsecond-order identities:

0 3

S1

S2 CCCWS3 A

AAUS4

1 20 3

U.Andrews, J.P.Chen, G.Bonik, R.W.Martin, A.T., Waveequation on one-dimensional fractals with spectraldecimation. preprint

4 ULYSSES ANDREWS, GRIGORY BONIK, JOE P. CHEN, RICHARD W. MARTIN, AND ALEXANDER TEPLYAEV

prop099

prop099 Proposition 2.5. We have that En+1(f, f) > En(f, f) for any function f , andEn+1(h, h) = En(h, h) = E(h, h)

for a harmonic h.prop16

prop16 Proposition 2.6. The Dirichlet (energy) form E on I is local and regular, and is self-similar inthe sense that

E(f, f) =∑

j=1,2,3

1rjE(fFj, fFj).

The domains of E and of the corresponding Laplacian ∆µ are contained and dense in the spaceof continuous functions.The µ–Laplacian ∆µ satisfies the following Gauss–Green (integration by parts) formula

E(f, f) = C

∫ 1

0

f∆µfdµ+ ff ′∣∣10.

where µ is a unique probability self-similar measure with weights m1, m2, m3, that is

µ =∑

j=1,2,3

mjµFj.

Also∆µf(x) = lim

n→∞

(1+ 2

pq

)n∆nf(x)

where the discrete Laplacians

∆nf(xk) =

f(xk)− pf(xk−1)− qf(xk+1)

orf(xk)− qf(xk−1)− pf(xk+1)

are defined as the generators of the nearest neighbor random walks on Vn with transitionalprobabilities p and q assigned according to the weights of the corresponding intervals.

Note that by definition p = m2

m1+m2, q = m1

m1+m2. The transitional probabilities p and q can be

assigned inductively as shown on FigurefigRWfigRW2.1.

t t- 1 1t t t t- - - 1

m1 m2 m3

q p p q 1t t t t t t t t t t- - - - - - - - - 1 q p p q q p q p p q p q q p p q 1

Figure 2.1. Random walks corresponding to the discrete Laplacians ∆n.figRWfigRW

Note: the above construction of the standard Laplacian and the associated Dirichlet form onI = [0, 1] corresponds to the case p = 1

2. In the case that p 6= 1

2then a change of variables can

turn the Dirichlet form into the standard one, or the measure into Lebesgue measure, but not

Resistance and measure weights:

r1 = r3 =p + pq

2 + pqand r2 =

2q − pq

2 + pq,

m1 = m3 =q

1 + qand m2 =

p

1 + q.

Up to a constant, the resistance weights are reciprocals of themeasure weights, and m1 + m2 + m3 = r1 + r2 + r3 = 1.

By the result of Kigami and Lapidus 1993, the Dirichlet orNeumann Laplacian ∆µ has the spectral asymptotics

0 < lim infλ→∞

ρ(λ)

λds/26 lim sup

λ→∞

ρ(λ)

λds/2<∞

where ρ(λ) is the eigenvalue counting function, and the spectraldimension is

ds =log 9

log(1+ 2

pq

) 6 1

where the inequality is strict if and only if p 6= q = 1− p.

We have spectral decimation, which makes big gaps in thespectrum, which implies better Fourier series convergence(Strichartz)

R(z)=z(z2−3z+2+pq)/pq

-

6 (2, 2)rr

r r rrIf p 6= 1

2 then Julia set of R(z) is a Cantor set of Lebesguemeasure zero.

Dirichlet kernel D82(x), n = 4, p = 0.8

0

0.5

1

0 0.2 0.4 0.6 0.8 1

Dirichlet kernel D82(x), n = 4, p = 0.2

0

0.5

1

0 0.2 0.4 0.6 0.8 1

Seehttp://homepages.uconn.edu/teplyaev/research/papers.html

http://homepages.uconn.edu/fractals/fractalwave/

for more information.

Waves propagate faster with increased fractality

In the next pictures we keep the same t = 0.1 and change spectraldimension from the classical case ds = 1, p = 0.5 towards muchmore fractal situation ds ≈ 0.7, p = 0.1

One can see that on a fractal waves propagate much faster (thetime was normalized so that the lowest non-zero eigenvalueλ1 = 1).

Wave propagation, p=0.5, t=0.1

Wave propagation, p=0.4, t=0.1

Wave propagation, p=0.3, t=0.1

Wave propagation, p=0.2, t=0.1

Wave propagation, p=0.1, t=0.1

In the next pictures we show t = 0.2 and, again, change spectraldimension from the classical case ds = 1, p = 0.5 towards muchmore fractal situation ds ≈ 0.7, p = 0.1, with the samenormalization λ1 = 1One can see that waves propagate faster with increased fractality(i.e. decreased spectral dimension in this case), although theprorogation is not linear in time.

Wave propagation, p=0.5, t=0.2

Wave propagation, p=0.4, t=0.2

Wave propagation, p=0.3, t=0.2

Wave propagation, p=0.2, t=0.2

Wave propagation, p=0.1, t=0.2