Abstracts of Invited Talks
Richard Bass, University of Connecticut
Uniqueness of Brownian Motion on the Sierpinski Carpet
The title pretty much says it all. In work with Martin Barlow, Takashi Kumagai, Sasha Teplyaev, and myself, we proved that, up to deterministic time change, there is only one symmetric non-degenerate strong Markov process with continuous paths whose state space is a (generalized) Sierpinski carpet and whose law is preserved under all the expected symmetries. A couple of corollaries: the construction of Barlow-Bass is scale invariant after all, and there is a well defined Laplacian on Sierpinski carpets. The morning talk will be an overview. For those wanting to see some of the details, Sasha and I will give one of the short courses in the afternoons.
David Croydon, University of Warwick
Local Limit Theorems for Fractal Graphs
Recently, there have been a number of weak scaling limit results proved for random walks on deterministic and random fractal graphs. In this talk, I will describe the conclusions of an article, co-authored with Ben Hambly, in which associated local limit theorems for the transition densities of these random walks are established. Our main theorem is a generalization of a result proved by Martin Barlow and Ben Hambly for supercritical percolation clusters, and in addition to describing this work, I will explain how our results apply to the homogenization problem for nested fractals, simple random walks on generalized Sierpinski carpet graphs and on graph trees converging to the continuum random tree.
Peter Grabner, Graz University of Technology
Poincaré Functional Equations, Harmonic Measures on Julia Sets, and Fractal Zeta Functions
Alexander Grigoryan, University of Bielefeld
The Dichotomy in the Heat Kernel Bounds on Fractals
Ben Hambly, University of Oxford
Masanori Hino, Kyoto University
Energy Measures and Derivatives on P.C.F. Self-Similar Sets
The martingale dimensions for a large class of p. c. f. self-similar sets are one. Using this result, it is shown that every function in the domain of the Dirichlet form has the "first-order derivative" almost everywhere with respect to the dominating energy measure, and its total energy is expressed as the square integral of the derivative. A few related topics will also be discussed.
John Hutchinson, Australian National University
V-Variable Fractals and their Analysis
Let F be a fixed class of IFSs acting on Rn. For each positive integer V there is a class of V variable fractals. These classes interpolate between the class of homogeneous random fractals corresponding to F (the V = 1 case) and the class of random recursive fractals corresponding to F (the V → ∞ case). The class of random V-variable fractals has a natural probability distribution. This class and its probability distribution can be approximated by a chaos game using a single deterministic IFS acting on the space of V-tuples of compact subsets of Rn.
I will first discuss the necessary background framework. (See the first three fractals research papers, also on the arXiv, but the informal survey paper there has been updated.) I will then discuss current work on analysis on V-variable fractals. The notion of a neck leading to spatial homogeneity at certain levels of magnification, and a Furstenberg Kesten type result for products of random V × V “flow” matrices, play important roles. This is joint work with Michael Barnsley and Örjan Stenflo, or with Uta Freiberg.
Jun Kigami, Kyoto University
Quasisymmetric Maps and Volume Doubling Measures
In this talk, we discuss relations between quasisymmetric maps and related notions, including volume doubling measures. We will also present how to construct a quasisymmetric distance which has desired property from a measure and an original distance. This talk can be considered as the second part of my lecture.
Ka-Sing Lau, Chinese University of Hong Kong
Cantor Boundary Behavior of Analytic Functions
Volodymyr Nekrashevych, Texas A&M University
Self-Similar Groups and Fractals
Hirofumi Osada, Kyushu University
Stochastic Differential Equations on the Sierpinski Carpet
I talk about stochastic differential equations on the plane and on the Sierpinski carpet. The coefficients of the SDEs are measurable functions and, in case of the carpet, contain Skolohod type terms. The solutions are given by the non-degenerate, self-similar diffusions on these spaces.
Anders Pelander, Narvik University College
Derivatives on p.c.f. Fractals and Product of Random Matrices.
We discuss how the local structure of smooth functions on p.c.f. fractals is related to properties of product of random matrices. This relation is used to define intrinsic first order derivatives for which a.e. differentiability w.r.t. self-similar measures can be shown for certain classes of functions and fractals using the Furstenberg-Kesten theory of products of random matrices. This gives an extension of the geography is destiny principle. (The local behavior at a point depends on the location of the point in the fractal.) The talk is on joint work with Alexander Teplyaev.
Luke Rogers, University of Connecticut
Traces of Sobolev Spaces to Domains in the Sierpinski Gasket
Bob Strichartz introduced Sobolev spaces on the Sierpinski Gasket (SG) and other p.c.f. fractals by the use of Bessel and Riesz potentials. Among other things, he showed that the trace of a function to a cell is in the same Sobolev space on the cell, and for integer orders of smoothness a function that is piecewise in the Sobolev spaces of cells and satisfies a finite number of matching conditions at junction points is in the global Sobolev space. The same matching conditions arise when one defines Sobolev spaces on domains as functions with some number of distributional Laplacian powers in a Lebesgue space, leading to the possibility that analogues of classical trace and extension theorems might hold in this context. This turns out not to be the case, essentially because the matching conditions are insufficient to determine the low-order behavior of functions once a significant number of cells are deleted. I will give an alternative set of matching conditions that can be used to describe the trace of the global Sobolev space on SG to a domain and explain why domains satisfying a measure density condition are those for which this trace has a bounded linear extension.
Robert Strichartz, Cornell University
Spectral Speculations Concerning Laplacians on Fractals, Graphs, and Manifolds
Laplacians may be defined on fractals, graphs and manifolds, and there are interesting connections among these 3 contexts. For example, Kigami defines Laplacians on certain fractals as limits of Laplacians on graphs. Colin de Verdiere shows that manifold Laplacians may be obtained similarly. One speculation I will discuss, called outer approximation, is that fractal Laplacians may sometimes be obtained as limits of ordinary Laplacians on planar domains. I will also speculate on connections between segments of the spectrum of a Laplacian and the geometry of the space at different scales. I will also discuss fractal analogs of elementary models in quantum mechanics (harmonic oscillator, hydrogen atom).
Alexander Teplyaev, University of Connecticut
Remarks on Spectral Decimation
It is well known that the spectrum of the Laplacian on the Sierpinski gasket can be computed using the so-called spectral decimation method. We show how to extend this approach to all finitely ramified fractals that have plenty of symmetries. We develop a matrix analysis, including analysis of singularities (so called exceptional values), which allows us to compute eigenvalues and eigenfunctions. The computations are straightforward and readily yield the multiplicity of eigenvalues. We consider such examples as the Sierpinski gasket, a non-p.c.f. analog of the Sierpinski gasket, the Level-3 Sierpinski gasket, a fractal 3-tree, the Hexagasket, the Diamond fractal, and one-dimensional examples. This is joint work with Neil Bajorin, Tao Chen, Alon Dagan, Catherine Emmons, Mona Hussein, Michael Khalil, Poorak Mody, and Benjamin Steinhurst.
Jeremy Tyson, University of Illinois
Dimension Comparison and Fractal Geometry in Carnot Groups
We find sharp comparison theorems relating Euclidean and Carnot-Carathéodory (CC) Hausdorff measures and dimensions on an arbitrary Carnot (nilpotent stratified Lie) group G. To show sharpness we construct sets of minimal CC dimension for fixed Euclidean dimension. Such sets are "horizontal": they follow the lowest possible layers in the sub-Riemannian decomposition of the tangent bundle of G. Typically these sets are fractal from the perspective of both Euclidean and CC geometry. As a consequence, we obtain exact dimension formulas for a class of invariant sets of nonlinear, nonconformal Euclidean IFS of polynomial type. Inspired by Falconer's work on a.s. dimensions of Euclidean self-affine fractals we show that CC self-similar fractals are a.s. horizontal.
This talk is an invitation for future interaction and collaboration from one branch of "analysis on fractals" to another. No prior knowledge of sub-Riemannian geometry will be assumed.
Martina Zaehle, University of Jena
Generalized Bessel and Riesz Potentials in Metric Measure Spaces
We consider metric measure spaces admitting a contractive strongly continuous semigroup (Tt)t≥0 of transformations with infinitesimal generator A. Bessel and Riesz type potentials associated with (Tt) are introduced by means of a completely monotone function f. They can be interpreted as the operators f(I – A) and f(–A), resp. The problem of invertibility is treated and the corresponding Bessel potential spaces are introduced. If f = 1/g for a Bernstein function g then the inverse operators are given by g(I – A) and g(–A), resp., determined by subordination.
Under some additional asssumptions we derive existence
of Bessel and Riesz kernels
and equivalent expressions in terms of the metric d and the function f. In particular, for α-sets with walk dimension β and $f(x) = x–σ/2 the Riesz kernel is equivalent to d(x,y)–(α – (β/2)σ). The results on the Bessel potentials can be applied to SPDE on fractals.