# 5th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals

June 11–15, 2014

## Abstracts of Talks

### Keynote Speakers

#### A non-pcf fractal which makes analysis easy

Analysis on fractals without pcf property faces the problem that self-similarity can rarely be used to simplify the investigation. We present a non-pcf example for which the Dirichlet problem can be easily solved. The definition of the Laplacian will be discussed in the manner of Strichartz's 2001 paper. The example is a modification of the octagasket.

#### Modulus of continuity for local times of random walks on graphs

In this talk, I present continuity results for the local times of random walks on graphs in terms of the resistance metric. I will also explain a particular application to the case when the graphs in questions satisfy the property of uniform volume doubling in the resistance metric, as is the case for various self-similar fractals.

#### Martin and Poisson Boundaries and Low-Pass Filters

We give a characterization of low-pass filters for multivariable scaling functions (associated with multivariable multiresolution analyses and wavelet sets) via a Markov process on the tree of finite words over the digit set corresponding to the dilation matrix used.

#### Differential operators and generalized trigonometric functions on fractal subsets of the real line

Spectral asymptotics of second order differential operators of the form $d/dm$ $d/dx$ on the real line are well known if $m$ is a self similar measure with compact support. We extend the results to some more general cases such as random fractal measures and self conformal measures. Moreover, we give a representation of the eigenfunctions as generalized trigonometric functions. The results were obtained in collaboration with Sabrina Kombrink and Peter Arzt.

#### Heat kernel and Harnack inequality

We show that the elliptic Harnack inequality follows from the volume doubling condition, Poincare inequality, and generalized capacity bound. Combining this with previous work, we obtain various equivalent conditions for heat kernel two-sided estimates.

#### Some spectral properties of pseudo-differential operators on the Sierpinski Gasket

We describe some results on the asymptotic behaviour of pseudo-differential operators on the Sierpinski gasket. We present the asymptotics of clusters of eigenvalues of generalized Schrodinger operators, generalizing some work due to Okoudjou and Strichartz. We also describe the asymptotics of the trace of continuous functions of pseudo-differential operators. Our results generalize the classical limit theorem of Szego and its extension to pseudo-differential operators on manifolds, and extend some recent work of Okoudjou, Rogers, and Strichartz. This presentation is based on joint work with Kasso Okoudjou and Luke Rogers.

#### Harmonic analysis on fractals

While we have in mind a wider variety of fractals than those defined by affine similarity mappings, the affine case will be our main focus. We will discuss three candidates for an harmonic analysis; (1) orthonormal bases in $L^2$ of the fractal under consideration; (2) a version making use of representation of a certain non-abelian algebra (generators and relations; and finally (3) one making use of energy Laplacians and boundaries. Each one has its advantages and limitations; both will be discussed. The talk will be based on joint research with several colleagues, especially, Dorin Dutkay, Erin Pearse, Steen Pedersen, Myung-Sin Song.

#### Simple random walk on the two-dimensional uniform spanning tree and the scaling limits

In this talk, we will first summarize known results about anomalous asymptotic behavior of simple random walk on the two-dimensional uniform spanning tree. We then show the existence of subsequential scaling limits for the random walk, and describe the limits as diffusions on the limiting random real trees embedded into Euclidean space. Anomalous heat transfer on the random real trees will be observed by estimating heat kernels of the diffusions. This is an on-going joint project with M.T. Barlow (UBC) and D. Croydon (Warwick).

#### Lipschitz equivalent of self-similar sets and hyperbolic graphs

Recently there is a lot of interest on the equivalence of Lipschtiz equivalence of self-similar sets. Our approach is to use the augmented tree structure (Kamaimovich) on the symbolic space, which is a hyperbolic graph in the sense of Gromov. We define a class of “simple” augmented tree on the IFS, and show that the associated self-similar sets are Lipschitz equivalent to the Cantor sets. This is a join work with Guotai Deng and Jason Luo.

#### Eigenvalue estimates of Laplacians defined by fractal measures

We study various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Euclidean spaces. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. By using properties of self-similar measures, such as Strichartz's second-order self-similar identities, we improve some of the eigenvalue estimates.

#### Uniqueness of eigenform on fractals

I present some recent results on the uniqueness of self-similar energies on finitely ramified fractals. In particular, some necessary and sufficient or sufficient conditions for nonuniqueness are given. These conditions amounts to the existence of two disjoint subsets of the set of the initial vertices that are stable under some special transformations. As a consequence of such conditions, there exist several fractals where the self-similar energy is not unique whose structures are very different from a tree.

#### Discrete homotopies and the fundamental group

Discrete homotopy methods were developed by Berestovskii, Plaut, and Wilkins over the last few years. They allow “stratification” of the fundamental group according to the “size of the holes” being measured. This allows study of spaces with bad local topology, such as fractals, but also allows additional understanding of the relationship between topology and geometry at a given scale. In this talk, I will discuss some applications: A “curvature-free” generalization of Cheeger's fundamental group finiteness theorem and a quantitative generalization of a theorem of Gromov on generators for the fundamental group of a compact Riemannian manifold. This is joint work with Jay Wilkins. If there is time, I will discuss a new topological invariant based on discrete homotopy methods that can distinguish between spaces that can't be distinguished by standard topological invariants.

#### Exact spectrum of the Laplacian on a domain in the Sierpinski gasket

For a certain domain $\Omega$ in the Sierpinski gasket $\mathcal{SG}$ whose boundary is a line segment, a complete description of the eigenvalues of the Laplacian, with an exact count of dimensions of eigenspaces, under the Dirichlet and Neumann boundary conditions is presented. The method developed in this paper is a weak version of the spectral decimation method due to Fukushima and Shima, since for a lot of “bad” eigenvalues the spectral decimation method can not be used directly. Let $\rho^0(x)$, $\rho^\Omega(x)$ be the eigenvalue counting functions of the Laplacian associated to $\mathcal{SG}$ and $\Omega$ respectively. We prove a comparison between $\rho^0(x)$ and $\rho^\Omega(x)$ says that $0\leq \rho^{0}(x)-\rho^\Omega(x)\leq C x^{\log2/\log 5}\log x$ for sufficiently large $x$ for some positive constant $C$. As a consequence, $\rho^\Omega(x)=g(\log x)x^{\log 3/\log 5}+O(x^{\log2/\log5}\log x)$ as $x\rightarrow\infty$, for some (right-continuous discontinuous) $\log 5$-periodic function $g:\mathbb{R}\rightarrow \mathbb{R}$ with $0<\inf_{\mathbb{R}}g<\sup_\mathbb{R}g<\infty$. Moreover, we explain that the asymptotic expansion of $\rho^\Omega(x)$ should admit a second term of the order $\log2/\log 5$, that becomes apparent from the experimental data. This is very analogous to the conjectures of Weyl and Berry.

#### Quantum gravity, asymptotic safety, and fractals

After a brief review of the asymptotic safety approach to quantum gravity we discuss the emergence of effective spacetimes which possess fractal properties present recent results on their scale dependent spectral dimension and use them for a detailed comparison of the statistical mechanics based and the continuum approach to asymptotically safe quantum gravity.

#### The “hot spots” conjecture on p.c.f. self-similar sets

In this talk, we will discuss the recent progresses on the “hot spots” conjecture of p.c.f. self-similar sets, including the standard Sierpinski gasket, the Sierpiski gasket with level 3, and Sierpinski gaskets in higher dimensional case.

#### Geometry and $\infty$-Poincare inequality vs. $p$-Poincare inequality

The focus of the talk will be on the geometric characterizations of doubling metric measure spaces supporting $\infty$-Poincare inequality, and the distinctions between $\infty$-Pooincare inequality and $p$-Poincare inequality. This talk is based on joint work with Estibalitz Durand-Cartagena, Jesus Jaramillo, and Alex Williams

#### Generalized Witt vectors and Lipschitz functions between Cantor sets

Burnside-Witt vectors form a ring, much like the $p$-adic integers, but they have a more complicated algebraic structure. In earlier work a collection of prime ideals of this ring were found. However it was also possible to show that this list of ideals was not complete. In this talk I will demonstrate where a second, very large, collection of prime ideals can be found in the ring of Burnside-Witt vectors and how they relate to continuous functions between two Cantor sets, which are given a $p$-adic algebraic structure. The result of the link to continuous functions is that it is now easy to find long chains of prime ideals and it is possible to show that the Krull dimension of the Burnside-Witt vectors is infinite.

#### Generalized Witt vectors and Lipschitz functions between Cantor sets

Burnside-Witt vectors form a ring, much like the $p$-adic integers, but they have a more complicated algebraic structure. In earlier work a collection of prime ideals of this ring were found. However it was also possible to show that this list of ideals was not complete. In this talk I will demonstrate where a second, very large, collection of prime ideals can be found in the ring of Burnside-Witt vectors and how they relate to continuous functions between two Cantor sets, which are given a $p$-adic algebraic structure. The result of the link to continuous functions is that it is now easy to find long chains of prime ideals and it is possible to show that the Krull dimension of the Burnside-Witt vectors is infinite.

#### Two way conversations between fractal analysis and classical analysis

Classical analysis speaks to fractal analysis in many ways, but it is not always a one way conversation. I will describe a few of these interactions, including two new works in which interesting results in classical analysis have emerged by listening to some things that fractal analysis has to say. The papers, “Graph paper trace charactereizations of functions of finite energy” and “Unexpected spectral asymptotics for wave equations on certain compact spacetimes” (with Jonathan Fox) have been posted on arXiv.

#### Waves, energy on fractals and related questions

I will present some recent results dealing with wave equation on one dimensional fractals (joint projects with J. F.-C. Chan, Sze-Man Ngai, Ulysses Andrews, Grigory Bonik, Joe P. Chen, Richard W. Martin). If time permits, existence and uniqueness of local energy forms on fractals will be discussed, as well as applications to vector valued PDEs on fractals.

### Other Speakers

#### Existence of resitance forms in some (non self similar) fractal spaces

We construct resistance forms in so called fractal quantum graphs and discuss other fractal spaces where such a construction is also possible. We also present some examples of associated Dirichlet forms in the fractal quantum graph case. This talk is based in joint work with D. Kelleher and A. Teplyaev.

#### Heat content asymptotics for some random Koch snowflakes

We consider the short time asymptotics of the heat content $E$ of a domain $D$ of $\mathbb{R}^d$. The novelty of this paper is that we consider the situation where $D$ is a domain whose boundary $\partial D$ is a random Koch type curve. When $\partial D$ is spatially homogeneous, we show that we can recover the lower and upper Minkowski dimensions of $\partial D$ from the short time behaviour of $E(s)$. Furthermore, in some situations where the Minkowski dimension exists, finer geometric fluctuations can be recovered and the heat content is controlled by $s^\alpha e^{f(\log(1/s))}$ for small $s$, for some $\alpha \in (0, \infty)$ and some regularly varying function $f$. The function $f$ is not constant is general and carries some geometric information. When $\partial D$ is statistically self-similar, then the Minkowski dimension and content of $\partial D$ typically exist and can be recovered from $E(s)$. Furthermore, the heat content has an almost sure expansion $E(s) = c s^{\alpha} N_\infty + o(s^\alpha)$ for smalls, for some $c$ and $\alpha \in (0, \infty)$ and some positive random variable $N_\infty$ with unit expectation arising as the limit of some martingale.

#### Asymptotics of cover times of random walks on fractal-like graphs

Random walks on fractal-like graphs have been an intensely studied subject over the past three decades. Much is known about the convergence of the rescaled random walks to a “Brownian motion” on the limit object, which relies upon, among other things, resistance estimates on the underlying graphs. In this talk, we will address the “cover time” problem of random walks on certain classes of fractal graphs. Recall that the cover time of a finite graph is the first time that a random walk covers all vertices of the graph. A theorem of Aldous (1991) states that along a sequence of increasing graphs, if the hitting times grow at a slower rate than the cover times, then the cover times exhibit exponential concentration toward the mean (namely, the fluctuations about the mean cover times are of lower order). Using the aforementioned resistance estimates, we obtain the asymptotics of the expected cover times on high-dimensional fractal graphs, such as the 3D Menger sponge graphs. This involves a nontrivial connection between cover times and the maxima of Gaussian free fields established by Ding, Lee, and Peres, as well as extensions of ideas used to prove entropic repulsion of the free field by Ugurcan and the speaker. If time permits we will also touch on the nature of the fluctuations.

#### Trace and extension results for a class of ramified domains with fractal self-similar boundary

We study some questions of analysis in view of the modeling of tree-like structures, such as the human lungs. More particularly, we focus on a class of planar ramified domains whose boundary contains a fractal self-similar part, noted $\Gamma$. We first study the Sobolev regularity of the traces on the fractal part $\Gamma$ of the boundary of functions in some Sobolev spaces of the ramified domains. Then, we study the existence of Sobolev extension operators for the ramified domains we consider. In particular, we show that there exists $p^*\in(1,\infty)$ such that there are $W^{1,p}$-extension operator for the ramified domains for every $1<p<p^*$, and the result is sharp. The construction we propose is based on a Haar wavelet decomposition on the fractal set $\Gamma$. Finally, we compare the notion of self-similar trace on the fractal part of the boundary with more classical definitions of trace.

#### Lattice approximation of attractors in the Hausdorff metric

Given a self-similar system in some Euclidean space, the associated scaling ratio vector is used to categorize the system as lattice or nonlattice. Under certain conditions it has been shown that this distinction will imply a great deal about the structure of the complex dimensions, which in turn will determine if a system is Minkowski measurable or not. The scope of this talk will be concerned with presenting a geometric aspect to the lattice/nonlattice dichotomy. In particular, given some Euclidean space, it will be shown that for any nonlattice self-similar system with attractor $F$ there exists a sequence of lattice self-similar systems with attractors $F_m$ such that the sequence $F_m$ converges to $F$ in the Hausdorff metric space. Furthermore, an additional result will be proven which will imply that the sequence of box counting functions generated by each of the $F_m$ converge pointwise to the box counting function of $F$.

#### Uniformization of planar Jordan domains

In the 90's, Stephenson conjectured that the Riemann mapping can be approximated by a sequence of finite networks. We will describe work in progress towards a resolution of a generalized version of his conjecture. The main ingredients include harmonic analysis on graphs, singular level curves of piecewise linear functions and Sobolev spaces of a fractional dimension.

#### Hardy inequalities in Triebel-Lizorkin spaces

In this talk I consider inequalities of Hardy type for functions in Triebel-Lizorkin spaces $F^s_{pq}(G)$ on a domain $G\subset \mathbb{R}^n$, whose boundary has the Aikawa dimension strictly less than $n-sp$. If $1<p<\infty$, in the class of domains whose boundary is compact and whose complement has zero Lebesgue measure, one could obtain a characterization for the validity of Hardy inequality in terms of the Aikawa dimension of $\partial G$.

The mentioned result applies in the case when the boundary $\partial G$ of a domain $G$ is ‘thin’. I would also like to discuss some of the known results for ‘fat’ sets. In particular, the validity of Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. In addition, I give a short exposition of various fatness conditions related to the theory, and apply Hardy inequalities in connection to the boundedness of extension operators for Triebel-Lizorkin spaces.

The talk is based on joint work with Antti Vähäkangas and joint work with Juha Lehrbäck, Heli Tuominen, and Antti Vähäkangas.

#### Intrinsic metrics and vector analysis for Dirichlet forms on fractals

We will discuss the possibility of defining vector analysis for measurable Dirichlet forms (quadratic forms on scalar functions). This vector analysis has applications to the Dirac operator, and the existence of the intrinsic metrics. This construction combines ideas from classical and non-commutative functional analysis and if time permits we shall discuss how this leads to the definition of spectral triples on fractal spaces.

#### Boundary Harnack principle on fractals

We prove a scale-invariant boundary Harnack principle (BHP) on inner uniform domains in metric measure spaces. We prove our result in the context of strictly local, regular, symmetric Dirichlet spaces, without assuming that the Dirichlet form induces a metric that generates the original topology on the metric space. Thus, we allow the underlying space to be fractal, e.g. the Sierpinski gasket.

#### Generalise $q$ dimension of self-affine set on Heisenberg group

We studied the generalized $q$-dimension of measures supported by Heisenberg self-affine sets in Heisenberg group, and we obtained bounds for it which are valid for ‘almost all’ classes of affine contractions. Especially, when $1<q\le 2$, exact values are established for Heisenberg self-affine measures.

#### Anderson localisation on the Sierpinski triangle

Filoche's and Mayboroda's theory demonstrate that both weak and Anderson localization originate from the same universal mechanism. This theory allows one to predict some localization properties, like the confining regions. We apply this theory to the case of the Sierpinski triangle and present our results.

#### Using Peano curves to contstruct Laplacians on fractals

We describe a new method to construct Laplacians on fractals using a Peano curve from the circle onto the fractal, extending an idea that has been used in the case of certain Julia sets. The Peano curve allows us to visualize eigenfunctions of the Laplacian by graphing the pullback to the circle. We study in detail three fractals: the pentagasket, the octagasket and the magic carpet. We also use the method for two nonfractal self-similar sets, the torus and the equilateral triangle, obtaining appealing new visualizations of eigenfunctions on the triangle. In contrast to the many familiar pictures of approximations to standard Peano curves, that do no show self-intersections, our descriptions of approximations to the Peano curves have self-intersections that play a vital role in constructing graph approximations to the fractal with explicit graph Laplacians that give the fractal Laplacian in the limit.

#### Differential equations on cubic Julia sets

The fractals REU in Cornell has successfully defined a Laplacian for several quadratic and cubic Julia sets, by approximating them as the limits of finite graphs and studying the subdivision rules governing the evolution of the graphs. This technique relied on the graphs subdividing in a nice way. However, complications arise for less well-behaved Julia sets. We will address how to resolve some of these complications by the methods of “procrastination” and taking limits, focusing specifically on cubic Julia sets. We will give specific examples of such Julia sets and analyze the spectra of their Laplacians.

#### Extensions and their minimizations on the Sierpinski gasket

In classical analysis one studies the finite variants of the Whitney extension theorem in which certain data is prescribed on a finite set of points in the Euclidean space and then extended by minimizing certain norms. In a similar vein, in this work, we study the extension problem on the Sierpinski Gasket ($SG$). We consider minimizing the functionals $\mathcal{E}_{\lambda}(f) = \mathcal{E}(f,f) + \lambda \int f^2 d \mu$ and $\int_{SG} |\Delta f(x)|^2 d \mu (x)$ with prescribed values at a finite set of points where $\mathcal{E}$ denotes the energy (the analog of $\int |\nabla f|^2$ in Euclidean space) and $\mu$ denotes the standard self-similiar measure on $SG$. In the first case, we explicitly construct the minimizer $f(x) = \sum_{i} c_i G_{\lambda}(x_i, x)$ for some constants $c_i$, where $G_{\lambda}$ is the resolvent for $\lambda \geq 0$. We minimize the energy over sets in $SG$ by calculating the explicit quadratic form $\mathcal{E}(f)$ of the minimizer $f$. We consider properties of this quadratic form for arbitrary sets and then analyze some specific sets. One such set we consider is the bottom row of a graph approximation of $SG$. For the bottom row, we describe both the quadratic form and a discretized form in terms of Haar functions. In both cases, we show the existence and uniqueness to the minimization problem and then study the fine properties of the unique minimizers. This is joint work with Pak-Hin Li, Nicholas Ryder and Robert S. Strichartz.

#### Fractal geometry and complex dimensions in metric measure spaces

While classical analysis dealt primarily with smooth spaces, much research has been done in the last half century on expanding the theory to the nonsmooth case. Metric Measure (MM) spaces are the natural setting for such analysis, and it is thus important to understand the geometry of subsets of such spaces. This talk will be an introductory survey, first of MM spaces that arise naturally in varying fields, and second an overview of the current theory of complex dimensions in both the one dimensional case and the more recent higher dimensional theory. This recent theory should naturally generalize to MM spaces (given an additional regularity condition), and we will show preliminary results in that direction.