Weak Uncertainty Principles on Fractals

We use the analytic tools such as the energy, and the Laplacians defined by Kigami for a class of post-critically finite (pcf) fractals which includes the Sierpinski gasket (SG), to establish some uncertainty relations for functions defined on these fractals. Although the existence of localized eigenfunctions on some of these fractals precludes an uncertainty principle in the vein of Heisenberg’s inequality, we prove in this article that a function that is localized in space must have high energy, and hence have high frequency components. We also extend our result to functions defined on products of pcf fractals, thereby obtaining an uncertainty principle on a particular type of non-pcf fractal.     

Mean Value Properties of Harmonic Functions on Sierpinski Gasket Type Fractals

In this paper, we establish an analogue of the classical mean value property for both the harmonic functions and some general functions in the domain of the Laplacian on the Sierpinski gasket. Furthermore, we extend the result to some other p.c.f. fractals with Dihedral-3 symmetry.     

“Graph paper” trace characterizations of functions of finite energy

We characterize functions of finite energy in the plane in terms of their traces on the lines that make up “graph paper” with squares of side length mⁿ for all n and certain 1/2-order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.     

Distribution theory on P.C.F. fractals

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.     

A trace theorem for the Dirichlet form on the Sierpinski gasket

In connection with the theory for Brownian motion on fractals, a corresponding Dirichlet form has been defined. We consider here the fractal known as the Sierpinski gasket, and characterize the trace of the domain of the Dirichlet form to the boundary of the gasket, boundary in this context meaning the triangle which confines the gasket.     

THE NAGUMO EQUATION ON SELF－SIMILAR FRACTAL SETS

The Nagumo equationut ut=△u+bu(u-a)(1-u),t＞0 is investigated with initial data and zero Neumann boundary conditions on post-critically finite (p.c.f.) self-similar fractals that have regular harmonic structures and satisfy the separation condition. Such a nonlinear diffusion equation has no travelling wave solutions because of the“pathological” property of the fractal. However, it is shown that a global Hoelder continuous solution in spatial variables exists on the fractal considered. The Sobolev-type inequality plays a crucial role, which holds on such a class of p.c.f self-similar fractals. The heat kernel has an eigenfunction expansion and is well-defined due to a Weyl‘s formula. The large time asymptotic behavior of the solution is discussed, and the solution tends exponentially to the equilibrium state of the Nagumo equation as time tends to infinity if b is small.     

Dirac and magnetic Schrödinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schrödinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Röckner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.     

Function spaces on fractals

We construct function spaces, analogs of Hölder-Zygmund, Besov and Sobolev spaces, on a class of post-critically finite self-similar fractals in general, and the Sierpinski gasket in particular, based on the Laplacian and effective resistance metric of Kigami. This theory is unrelated to the usual embeddings of these fractals in Euclidean space, and so our spaces are distinct from the function spaces of Jonsson and Wallin, although there are some coincidences for small orders of smoothness. We show that the Laplacian acts as one would expect an elliptic pseudodifferential operator of order d+1 on a space of dimension d to act, where d is determined by the growth rate of the measure of metric balls. We establish some Sobolev embedding theorems and some results on complex interpolation on these spaces.     

p-Energy and p-Harmonic Functions on Sierpinski Gasket Type Fractals

We show that it is possible to define a notion of p-energy for functions defined on a class of fractals including the Sierpinski gasket (SG) for any value of p, 1<p<, extending the construction of Kigami for p=2, as a renormalized limit of modified p-energies on a sequence of graphs. Our proof is non-constructive, and does not settle the question of uniqueness. Based on the p-energy we may define p-harmonic functions as p-energy minimizers subject to boundary conditions, but again uniqueness is only conjectural. We present some numerical data as a complement to our results. This work is intended to pave the way for an eventual theory of p-Laplacians on fractals.     

Multiple solutions for a class of nonlinear elliptic equations on the Sierpinski gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of △u+ c(x)u = f(x, u), with zero Dirichlet boundary conditions on the Sierpihski gasket. Our existence results do not require any growth conditions of f(x,t) in t, in contrast to the classical theory of elliptic equations on smooth domains.     

Criteria for Spectral Gaps of Laplacians on Fractals

Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove general criteria for the existence of gaps when the Laplacian admits spectral decimation. The known examples, including the Sierpinski gasket and the level-3 Sierpinski gasket, and the new examples including the fractal-3 tree, the Hexagasket and the infinite family of tree-like fractals satisfy the criteria.     

Products of random matrices and derivatives on p.c.f. fractals

We define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere with respect to self-similar measures for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle to these cases, and also obtain results on the pointwise behavior of local eccentricities on the Sierpiński gasket, previously studied by Öberg, Strichartz and Yingst, and the authors. We also establish the relation of the derivatives to the tangents and gradients previously studied by Strichartz and the authors. Our main tool is the Furstenberg-Kesten theory of products of random matrices.     

一类含参变量的Sierpinski垫片的Hausdorff测度

Sierpinski垫片是具有严格自相似性的经典分形集之一.本文给出了一类含参变量的Sierpinski垫片.通过它在x轴上的投影估计了这类Sierpinski垫片的Hausdorff测度的下界,然后精心构造了一个仿射变换,将参变量的范围由(0,π/3)的讨论转换到(π/3,π)的讨论,从而得到了这类Sierpinski垫片的Hausdorff测度的精确值.     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

Derivations and Dirichlet forms on fractals

We study derivations and Fredholm modules on metric spaces with a local regular conservative Dirichlet form. In particular, on finitely ramified fractals, we show that there is a non-trivial Fredholm module if and only if the fractal is not a tree (i.e. not simply connected). This result relates Fredholm modules and topology, refines and improves known results on p.c.f. fractals. We also discuss weakly summable Fredholm modules and the Dixmier trace in the cases of some finitely and infinitely ramified fractals (including non-self-similar fractals) if the so-called spectral dimension is less than 2. In the finitely ramified self-similar case we relate the p-summability question with estimates of the Lyapunov exponents for harmonic functions and the behavior of the pressure function.     

Transition Density Estimates for Diffusion Processes on Post Critically Finite Self-Similar Fractals

The framework of post critically finite (p.c.f) self-similarfractals was introduced to capture the idea of a finitely ramifiedfractal, that is, a connected fractal set where any componentcan be disconnected by the removal of a finite number of points.These ramification points provide a sequence of graphs whichapproximate the fractal and allow a Laplace operator to be constructedas a suitable limit of discrete graph Laplacians. In this paperwe obtain estimates on the heat kernel associated with the Laplacianon the fractal which are best possible up to constants. Theseare short time estimates for the Laplacian with respect to anatural measure and expressed in terms of an effective resistancemetric. Previous results on fractals with spatial symmetry haveobtained heat kernel estimates of a non-Gaussian form but whichare of Aronson type. By considering a range of examples whichare not spatially symmetric, we show that uniform Aronson typeestimates do not hold in general on fractals. 1991 MathematicsSubject Classification: 60J60, 60J25, 28A80, 31C25.     

Some properties of fractal interpolation functions on Sierpinski gasket

In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min-max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed.     

Dirac operators and spectral triples for some fractal sets built on curves

We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space. Several fractals, like a finitely summable infinite tree and the Sierpinski gasket, fit naturally within our framework. In these cases, we show that our spectral triples do describe the geodesic distance and the Minkowski dimension as well as, more generally, the complex fractal dimensions of the space. Furthermore, in the case of the Sierpinski gasket, the associated Dixmier-type trace coincides with the normalized Hausdorff measure of dimension log3/log2.     

Spectral zeta functions of fractals and the complex dynamics of polynomials

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.

     

Some asymptotic results for the combination of evidence problem

The combination of evidence problem is treated here as the construction of a posterior possibility function (or probability function, as a special case) describing an unknown state parameter vector of interest. This function exhibits the appropriate components contributing to knowledge of the parameter, including conditions or inference rules, relating the parameter with observable characteristics or attributes, and errors or confidences of observed or reported data. Multivalued logic operators - in particular, disjunction, conjunction, and implication operators, where needed – are used to connect these components and structure the posterior function. Typically, these operators are well-defined for only a finite number of arguments. Yet, often in the problem at hand, a number of observable attributes represent probabilistic concepts in the form of probability density functions. This occur, for example, for attributes representing ordinary numerical measurements- as opposed to those attributes representing linguistic-based information, where non-probabilistic possibility functions are used. Thus the problem of discretization of probabilistic attributes arises, where p.d.f.'s are truncated and discretized to probability functions. As the discretization process becomes finer and finer, intuitively the posterior function should better and better represent the information available. Hence, the basic question that arises is: what is the limiting behavior of the resulting posterior functions when the level of discretization becomes infinitely fine, and, in effect, the entire p.d.f.'s are used?It is shown in this paper that under mild analytic conditions placed upon the relevant functions and operators involved, nontrivial limits in the above sense do exist and involve monotone transforms of statistical expectations of functions of random variable corresponding to the p.d.f.'s for the probabilistic attributes.