# 3rd Conference on Analysis and Probability on Fractals

June 11–15, 2008

## Abstracts of Contributed Talks

### Approximation of Gaussian Random Fields: General Results and Optimal Wavelet Representation of the Lévy Fractional Motion

We investigate the approximation rate for certain centered Gaussian fields by a general approach. Upper estimates are proved in the context of so-called Hölder operators and lower estimates follow from the eigenvalue behavior of some related self-adjoint integral operator in a suitable L2(μ)-space. In particular, we determine the approximation rate for the Lévy fractional Brownian motion XH with Hurst parameter H ∈ (0,1), indexed by a self-similar set T ⊂ RN of Hausdorff dimension D. This rate turns out to be of order nH/D(log n)1/2. In the case T = [0,1]N we present a concrete wavelet representation of XH leading to an approximation of XH with the optimal rate nH/N(log n)1/2.

### Eigenvalues and Eigenfunctions of the Laplacian Defined by the Infinite Bernoulli Convolution Associated with the Golden Ratio

We use the finite element method to obtain numerical approximations to the eigenfunctions and eigenvalues of Laplacian defined by the infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Canton measure. The iterated function systems defining these measures do not satisfy open set condition; We use second-order self-similar identities introduced by Strichatz et al. to analyse the measures. This is a joint work with Sze-Man Ngai.

### Spectral Theory for the Fractal Laplacian in the Context of h-Sets

Let h : (0,1] → R+ denote a continuous monotone increasing function such that h(0+) = 0. Assume that there exist a non-empty compact set Γ ⊂ Rn and a finite Radon measure μ such that

supp μ = Γ and μ(B(γ, r)) ∼ h(r),  0 < r ≤ 1, γ ∈ Γ.

We say that such a set Γ is an h-set. Let Ω be a bounded C domain in Rn with Γ ⊂ Ω. We study the operator

:= (–Δ)–1 ° tr Γ,

acting in convenient function spaces in Ω, where (–Δ)–1 is the inverse of the Dirichlet Laplacian in Ω and

tr Γ = id Γ ° tr Γ.

We present estimations for the eigenvalues of B and discuss the smoothness of the associated eigenfunctions. We remark that the particular class of d-sets was studied by Triebel and the case of (d, ψ)-sets was studied by Edmunds, Moura and Triebel. As it was mentioned in these works, the operator B has physical relevance, at least in the case n = 2: it describes the vibration of a drum where the whole mass is distributed on Γ.

### The Spectrum of the Energy Laplacian on the Sierpinski Gasket

We investigate the behavior of the eigenvalues and eigenfunctions of the energy Laplacian δν on the Sierpinski Gasket (SG) subject to the Dirichlet and Neumann boundary conditions, using the finite element method and a method that involves discretizing the pointwise formula for δν. From our numerical computations we observe that the eigenvalue counting function obeys the asymptotic behavior predicted by Weyl's theorem, and we conjecture that the limit of the Weyl ratio exists, which is not the case for the standard Laplacian δμ on SG. We also conjecture that the multiplicities of the eigenvalues are either 1 or 2, in stark contrast to the high multiplicities of eigenvalues of δμ. Moreover, we note the existence of eigenfunctions that satisfy both Dirichlet and Neumann boundary conditions, and the relationship between the Kusuoka measure and the heat kernel estimate.

### Groebner Basis, Resultants and the Generalized Mandelbrot Set

This paper demonstrates how the Groebner Basis Algorithm can be used for finding the bifurcation points in the generalized Mandelbrot set. It also shows how resultants can be used to find components of the generalized Mandelbrot set. In addition, the numerical implementation is done with Maple V and some typical computational results will be
shown.

### Conformal Graph Directed Markov Systems

We present the main concepts and results for Graph Directed Markov Systems, particularly th ose systems that have a finitely irreducible incidence matrix. We show how these results may change when the incidence matrix is assumed to irreducible but not finitely irreducible. Several examples are given. Then we examine finite systems which are not irreducible. We derive necessary and sufficient conditions in terms of the strongly connected components of the directed graph in order that the Hausdorff measure of the limit set be positive and finite and show that if this condition doesn't hold, then the measure is infinite but σ finite. We provide a number of examples.

### Laplacians on Random Carpets

We will present a new method to approximate the Neumann spectrum of a Laplacian on a fractal K in the plane as a renormalized limit of the Neumann spectra of the standard Laplacian on a sequence of domains that approximate K from the outside. The method allows a numerical approximation of eigenvalues and eigenfunctions for lower portions of the spectrum. We present experimental evidence that the method works by looking at examples where the spectrum of the fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)). We then analyze the method's application to random variants of the Sierpinski Carpet (SC). At present we have no explanation as to why the method should work.

### Pseudodifferential Operations on Fractals

In this talk, which is based on joint work with Robert Strichartz, we present a theory of pseudodifferential operators defined on fractafolds. We extend some previous results due to Strichartz, Rogers, Teplyaev, Allan, Barany, and others. In the contstant coefficient case we show that these operators are bounded on Lp spaces. This fact allows us to define Sobolev spaces on fractafolds and prove a Sobolev embedding theorem. We also show that the symbolic calculus holds in this case and that a large class of these pseudodifferential operators satisfy the so called pseudo-local property. In the variable coefficient case we present a version of the Calderon-Vaillancourt Theorem.

### Self-Affine Sets of Kakeya Type

This is a joint work with P. Shmerkin. We compute the Minkowski dimension for a family of self-affine sets on R2. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do
not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.

### Weyl Type Spectral Asymptotics for the Laplacian on Sierpinski Carpets

Let {λn}n ≥ 1 be the eigenvalues of the Laplacian associated with the Brownian motion on a (possibly higher dimensional) Sierpinski carpet, and let T(t) := Σn = 1 e–λ_nt, > 0 (called the partition function of the Laplacian). B. M. Hambly, Asymptotics for functions associated with heat flow on the Sierpinski carpet (preprint 2008) has shown that there exists a (0,∞)-valued periodic continuous function G such that, as t\downarrow 0\$,

T(t) – td_S/2G(log t–1) = o(td_S/2).

In this talk I will present the following two results closely related to Hambly's result above:

1. T(t) – td_S/2G(log t–1) in the above formula also admits an asymptotic behavior of similar form (giving an affirmative answer to Hambly's conjecture in the above preprint).
2. Even if we consider a time change (with respect to a self-similar measure) of the original Brownian motion on the carpet, the associated partition function admits a similar asymptotic behavior as long as the transition density is subject to the Sub-Gaussian upper bound.

### Finite Type Iterated Functions Systems: a Computational Perspective

The finite type condition was presented by Wang and Ngai as a separation condition on an IFS that is weaker than the open set condition and yet still allows the computation of the Hausdorff dimension of the attractor. This has been further generalized by Ngai, Das, and Lau. These techniques do not yield a single simple formula, but an algorithm to generate a matrix; the dimension may then be computed in terms of the spectral radius of this matrix. In this talk we present a program, written in Mathematica, that fully implements the algorithm and allows the easy computation of the Hausdorff dimension of many examples. We also discuss a few computational issues that arise in the implementation.

### Analysis on Fractals Defined by Iterated Function Systems with Overlaps

We study spectral asymptotics of Laplacians defined by iterated function systems with overlaps in higehr-dimensions. Previously, this problem has been studied in one dimension. The problem is more difficult in higher dimensions. We will focus on a two-dimensional example.

### Towards a Countable Collection Of Periodic Orbits of the Koch Snowflake Billiard

We generalize the results of an equilateral triangle billiard to KSn, the nth level approximation of the Koch Snowflake. We give a proof that a billiard ball with initial condition (x0, nπ/3) with x0 the midpoint of a deleted segment of a pre-Cantor set (as found in KSn) will have a periodic trajectory. These particular trajectories are generated by a system of IFS's (iterated function systems) each acting on a different collection of points. We then give a supporting argument as to why our IFS-like function must then converge to an orbit of the Koch Snowflake billiard.

Other initial conditions are investigated and parallels with the equilateral triangle are discussed. Extending the tiling method, we propose that a fractal tiling of the plane may provide an alternate method of investigation.

### Operator Theory of Electrical Resistance Networks

An electrical resistance network (ERN) is a graph with edges weighted by a function whose values are interpreted as resistances; the resulting voltage drop between two vertices gives an intrinsic metric on the network. One may embed this network in a Hilbert space and use the resulting framework to understand various function spaces associated to the network. I will discuss how the presence of harmonic functions affects the relationship between the Dirichlet energy form and the (discrete) Laplace operator, and how this may be understood in terms of a certain notion of boundary on infinite graphs. The structure of the Hilbert space allows for easy solutions to some potential-theoretic problems and reveals the asymptotic geometry of the network.

### Existence of Self-Similar Energies of Fractals

It is conjectured that on any PCF self-similar set there exists a self similar energy, or in other words an eigenform for the renormalization operator with suitable weights. This was proved to be the case for fractals with three vertices or in some specific situations. In this talk, I describe a relatively general class of fractals where a self-similar energy does exist. However, in some of these fractals the eigenform found by this method is not regular.

### Covering Maps and Periodic Functions on Higher-Dimensional Sierpinski Gaskets

We construct covering maps from infinite blowups of the n-dimensional Sierpinski gasket SGn to certain compact fractafolds based on SGn. These maps are fractal analogs of the usual covering maps from the line to the circle. The construction extends work of the second author in the case n = 2, but a different method of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

### A Dirichlet Forms and Markov Processes on Laakso and Barlow-Evans Fractals of Arbitrary Dimension,

We consider two construction each of which we show can produce the same family of fractals. The Laakso construction provides minimal upper gradients and the associated Cheeger Sobolev space. The Barlow-Evans construction provides for a large collection of Markov processes on these fractals. We show explicitly which Dirichlet form is related to the minimal upper gradients and to which Markov process it is associated.

### On the Projections of Purely Unrectifiable One-Dimensional Hausdorff Measures

We will give a necessary and sufficient condition for a measure on the real-line to be an orthogonal projection of one-dimensional Hausdorff measure restricted to some purely 1-unrectifiable planar set. (This is joint work with M. Csörnyei from University College, London)