Uniqueness of eigenforms on fractals – II

I give an explicitly verifiable necessary and sufficient condition for the uniqueness of the eigenform on finitely ramified fractals, once an eigenform is known. This improves the results of my previous paper [16], where I gave some necessary and some sufficient conditions, and with a relatively mild additional requirement on the known eigenform. The result of this paper can be interpreted as a uniqueness result for self‐similar energies on finitely ramified fractals.     

Piecewise linear wavelets on sierpinski gasket type fractals

Let SG denote the Sierpinski gasket with Hausdorff measure μ of dimensionlog 3/log 2, let PL_k denote the continuous piecewise linear functions with respect to the usual triangulation of SG into 3^k triangles, and let W_k denote the orthogonal complement of PL_k−1 in PL_k. We construct a basis for each W_k, so that the entire collection is a frame for L²(dμ). This wavelet basis is obtained from three wavelet generators by scaling, translation and rotation, and the wavelets are supported either by one corner triangle or a pair of adjacent triangles in the triangulation of level k − 1. Analogous bases are constructed in the von Koch curve, the hexagasket, and the n-dimensional analog of SG.     

Homogenization on nested fractals

Summary We study the homogenization problem on nested fractals. LetX _t be the continuous time Markov chain on the pre-nested fractal given by puttingi.i.d. random resistors on each cell. It is proved that under some conditions, converges in law to a constant time change of the Brownian motion on the fractal asn, where is the contraction rate andt _E is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.Dedicated to Professor Masatoshi Fukushima on his 60th birthdayResearch partially supported by the Yukawa Foundation     

Approximation of infinite dimensional fractals generated by integral equations

We present some results concerning fractals generated by an iterated function system in the infinite dimensional space of continuous functions on a compact interval. Namely, we approximate the fractal via a finite approximant set and project this approximant set in two dimensions, in order to “draw” a picture of it.     

Semilinear evolution transmission problems across fractal layers

A semilinear parabolic transmission problem across either a fractal layer S$S$ or the corresponding prefractal layer _Sh$S_h$ is studied. Local existence, uniqueness and regularity results for the mild solution, in both cases, are established as well as a sufficient condition on the initial datum for global existence is given. The asymptotic behaviour of the solutions of the approximating problems is studied.     

The short-cut test

The short-cut test detects existence and uniqueness of “Laplacians” on finitely ramified, graph-directed fractals. Previous results by Sabot, Nussbaum and the author are improved and extended. It opens up the way for further studies because it combines well established spectral, dynamical and analytic techniques. Its algorithmic and recursive structure is designed to provide computable and flexible criteria.     

A trace theorem for Dirichlet forms on fractals

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on Sierpinski gaskets and Sierpinski carpets to their boundaries, where the boundaries are represented by triangles and squares that confine the gaskets and the carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields. These processes behave as an appropriate fractal diffusion within each fractal component of the field.     

Self-similarity and spectral asymptotics for the continuum random tree

We use the random self-similarity of the continuum random tree to show that it is homeomorphic to a post-critically finite self-similar fractal equipped with a random self-similar metric. As an application, we determine the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on the continuum random tree. We also obtain short time asymptotics for the trace of the heat semigroup and the annealed on-diagonal heat kernel associated with this Dirichlet form.     

The zeta function of the Laplacian on certain fractals

We prove that the zeta function of the Laplacian on self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.

     

Dimension spectra of random subfractals of self-similar fractals

The dimension of a point x   in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}$\{x\}$) is the algorithmic information density of x  . Roughly speaking, this is the least real number dim(x)$\dim (x)$ such that r×dim(x)$r\times \dim (x)$ bits suffice to specify x   on a general-purpose computer with arbitrarily high precision 2^−r$2^{-r}$. The dimension spectrum of a set X   in Euclidean space is the subset of [0,n]$[0,n]$ consisting of the dimensions of all points in X.     

NOTE ON A BOUNDARY VALUE PROBLEM OF POISSON＇S EQUATION

This note shows that fur a BVP of Poisson‘s equation with Qm(u,v) as its source function, a direct evaluation of some integrals yields the same exact results as obtained by similarity analysis.     

Morse-Sard定理在无限维时的一个反例

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｉｓａｆｕｎｄａｍｅｎｔａｌｔｈｅｏｒｅｍｉｎａｎａｌｙｓｉｓ ,ｅｓｐｅｃｉａｌｌｙｉｎｔｈｅｂａｓｉｓｏｆｔｒａｎｓｖｅｒｓａｌｉｔｙｔｈｅｏｒｙａｎｄｄｉｆｆｅｒｅｎｔｉａｌｔｏｐｏｌｏｇｙ .ＴｈｅｃｌａｓｓｉｃａｌＭｏｒｓｅ Ｓａｒｄｔｈｅｏｒｅｍｓｔａｔｅｓｔｈａｔｔｈｅｉｍａｇｅｏｆｔｈｅｓｅｔｏｆｃｒｉｔｉｃａｌｐｏｉｎｔｓｏｆａｆｕｎｃｔｉｏｎ ｆ :Ｒｍ→ＲｌｏｆｃｌａｓｓＣｍ -ｌ+1ｈａｓｚｅｒｏＬｅｂｅｓｇｕｅｍ…     

A fixed point theorem for moment matrices of self-similar measures

We consider the self-similar measure on the complex plane C$\mathbb{C}$ associated to an iterated function system (IFS) with probabilities. From this IFS we define an operator in a complete metric space of infinite matrices. Using the expression obtained in a previous work of the authors, we prove that this operator has as fixed point the moment matrix of the self-similar measure. As a consequence, we obtain a very efficient algorithm to compute the moment matrix of the self-similar measure. Finally, in order to estimate the rate of convergence of the algorithm, we find an upper bound of the norm of this contractive operator.     

The singularity spectrum of Lévy processes in multifractal time

Let X=(Xt)_t?0 be a Lévy process and μ a positive Borel measure on R₊. Suppose that the integral of μ defines a continuous increasing multifractal time . Under suitable assumptions on μ, we compute the singularity spectrum of the sample paths of the process X in time μ defined as the process (XF(t))_t?0.A fundamental example consists in taking a measure μ equal to an “independent random cascade” and (independently of μ) a suitable stable Lévy process X. Then the associated process X in time μ is naturally related to the so-called fixed points of the smoothing transformation in interacting particles systems.Our results rely on recent heterogeneous ubiquity theorems.     

Approximation and Uniqueness for -Harmonic Functions

 In this paper, we consider the class of functions defined on the unit ball which are harmonic with respect to a certain class of degenerate elliptic operators. We establish some approximation and uniqueness theorems for these functions. Our work generalizes some recent results obtained by J. Bruna and K. T. Hahn in the particular case of the Laplace-Beltrami operator of the Bergman metric. (Received 17 September 1998)     

Some calculations on the conditional densities of well-admissible measures on linear spaces

When a probability measurem on a topological vector spaceE is well-admissible in a directionk E, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.     

On function spaces related to fractional diffusions on d-sets

We prove that all the Dirichlet forms associated with certain diffusions on a d-set are equivalent and that their common domain is an integral Lipschitz space. We also provide an analytic characterisation of the walk dimension d_w of a d-set F and show that all fractional diffusions on F share d_w as their walk dimension.     

On the exact rate of convergence of frequencies of digits in self-similar sets

We study the exact rate of convergence of frequencies of digits of “normal” points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of ?^d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in ?^d—namely, sets with full Hausdorff dimension—for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.