The Hausdorff measure of the self-similar sets

The self-similar sets satisfying the open condition have been studied. An estimation of fractal, by the definition can only give the upper limit of its Hausdorff measure. So to judge if such an upper limit is its exact value or not is important. A negative criterion has been given. As a consequence, the Marion’s conjecture on the Hausdorff measure of the Koch curve has been proved invalid. Project partially supported by the State Scientific Commission and the State Education Commission.     

A dimensional result for random self-similar sets

A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal

     

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

Hausdorff dimension of Cantor sets and polynomial hulls

We give examples of Cantor sets in of Hausdorff dimension 1 whose polynomial hulls have non-empty interior.

     

Beurling dimension and self-similar measures

In this paper the authors study the Beurling dimension of Bessel sets and frame spectra of some self-similar measures on ${\mathbb{R}}^d$ and obtain their exact upper bound of the dimensions, which is the same given by Dutkay et al. (2011) [8]. The upper bound is attained in usual cases and some examples are given to explain our theory.     

自相似集的质量分布原理与Hausdorff测度及其应用

本文提出了满足开集条件的自相似集的质量分布原理.作为应用,得到了计算一类满足开集条件的自相似集的Hausdorff测度的准确值的方法,并举例说明了此方法对于计算一类满足开集条件的自相似集的Hausdorff测度的准确值是行之有效的.     

Some necessary and sufficient conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion. This paper is a part of the second author’s Doctoral thesis, written at Shiraz University under the direction of the first author.     

The stability of Hausdorff dimension for the level sets under the perturbation of conformal repellers

Let $M$ be a $C^\infty$ compact Riemann manifold. $f:M\to M$ is a $C^1$ map and $\Lambda_f \subset M$ is a conformal repeller of $f$. Suppose $\varphi:M\to\mathbb{R}$ is a continuous function and let $f_k$ be nonconformal perturbation of the map $f$. We consider the stability of Hausdorff dimension of level sets for Birkhorff average of potential function $\varphi$ with respect to $f_k$ and $f$.     

The nonmonotonic twist maps and basic hyperbolic sets

The existence of horseshoes for a family of nonmonotonic twist maps is established. And it is proved that the Hausdorff dimension of the horseshoes goes to the dimension of phase space as the parameter goes to infinity.     

Upper and lower Hausdorff dimension estimates for invariant sets of weak k‐1‐endomorphisms

For a special class of non‐injective maps called the weak k‐1‐endomorphisms on Riemannian manifolds upper and lower bounds for Hausdorff dimensions of invariant sets are given in terms of the singular values of the tangent maps, which generalize Franz's corresponding results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

The dimension of the kernel in finite and infinite intersections of starshaped sets

Summary. We establish the following Helly-type result for infinite families of starshaped sets in Define the function f on {1, 2} by f(1) = 4, f(2) = 3. Let be a fixed positive number, and let be a uniformly bounded family of compact sets in the plane. For k = 1, 2, if every f(k) (not necessarily distinct) members of intersect in a starshaped set whose kernel contains a k-dimensional neighborhood of radius , then is a starshaped set whose kernel is at least k-dimensional. The number f(k) is best in each case. In addition, we present a few results concerning the dimension of the kernel in an intersection of starshaped sets in Some of these involve finite families of sets, while others involve infinite families and make use of the Hausdorff metric.     

Energy measures on p.c.f. self-similar sets

On connected post critically finite (p.c.f.) self-similar sets we give a linear extension method to compute the energy measures of harmonic functions with respect to the standard energy, and as an application we also compute the L²$L^2$ dimensions of these measures on some p.c.f. self-similar sets.     

The necessary condition and sufficient conditions for wavelet frame on local fields

In the paper entitled “Multiresolution analysis on local fields” [H.K. Jiang, D.F. Li, N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294 (2) (2004) 523-532], we establish the orthonormal wavelet construction from multiresolution analysis on local fields. The objective of this paper is to construct wavelet frame on local fields. A necessary condition and four sufficient conditions for wavelet frame on local fields are given. An example is presented at the end.     

The necessary and sufficient conditions for dependent quadratic forms to be distributed as multivariate gamma

Let S be distributed as noncentral Wishart given by $\text{W}{}_{\mathrm{p}}\text{(m, Σ, Ω)}$ and let x be an n × 1 random vector distributed as N(μ, V). If q_i = x′A_ix + 2l′_ix + c_i, i = 1, 2,…, p, are p dependent second degree polynomials in the elements of x where A_j's are symmetric matrices, then the necessary and sufficient conditions for q₁ , q₂ ,…, q_p to be distributed as the diagonal elements of S are established and this generalizes the result for Σ = I. Some special cases are considered.     

The necessary and sufficient conditions for the existence of periodic orbits in a Lotka-Volterra system

By extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka-Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits.