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.     

Arithmetic progressions in self-similar sets

Given a sequence {\{b_i\}}_{i=\mathrm{1}}^n and a ratio \lambda \in (\mathrm{0},\mathrm{1}), let E={\cup }_{i=\mathrm{1}}^n(\lambda E+b_i) be a homogeneous self-similar set. In this paper, we study the existence and maximal length of arithmetic progressions in E: Our main idea is from the multiple β-expansions.     

Embeddings of self-similar ultrametric Cantor sets

We study self-similar ultrametric Cantor sets arising from stationary Bratteli diagrams. We prove that such a Cantor set C is bi-Lipschitz embeddable in R[Hdim(C)]+1, where [Hdim(C)] denotes the integer part of its Hausdorff dimension. We compute this Hausdorff dimension explicitly and show that it is the abscissa of convergence of a zeta-function associated with a natural sequence of refining coverings of C (given by the Bratteli diagram). As a corollary we prove that the transversal of a (primitive) substitution tiling of Rd is bi-Lipschitz embeddable in Rd+1.We also show that C is bi-Hölder embeddable in the real line. The image of C in R turns out to be the ω-spectrum (the limit points of the set of eigenvalues) of a Laplacian on C introduced by Pearson-Bellissard via noncommutative geometry.     

On the open set condition for self-similar fractals

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

     

On singularity of energy measures on self-similar sets

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60)     

Multifractal formalism for self-similar measures with weak separation condition

For any self-similar measure μ on satisfying the weak separation condition, we show that there exists an open ball U₀ with μ(U₀)>0 such that the distribution of μ, restricted on U₀, is controlled by the products of a family of non-negative matrices, and hence μ|_U0 satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for μ|_U0 is valid on the whole range of dimension spectrum, regardless of whether there are phase transitions. Moreover the dimension spectra of μ and μ|_U0 coincide for q0. This result unifies and improves many of the recent works on the multifractal structure of self-similar measures with overlaps.     

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.     

The multifractal spectrum of statistically self-similar measures

We calculate the multifractal spectrum of a random measure constructed using a statistically self-similar process. We show that with probability one there is a multifractal decomposition analogous to that in the deterministic self-similar case, with the exponents given by the solution of an expectation equation.     

Quasisymmetric equivalence of self-similar sets

We prove that the self-similar sets with the strong separation condition are all quasisymmetrically equivalent.     

Weak chain-completeness and fixed point property for pseudo-ordered sets

In this paper the notion of weak chain-completeness is introduced for pseudo-ordered sets as an extension of the notion of chain-completeness of posets (see [3]) and it is shown that every isotone map of a weakly chain-complete pseudo-ordered set into itself has a least fixed point.     

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439-3444].     

The necessary and sufficient conditions for various self-similar sets and their dimension

We have given several necessary and sufficient conditions for statistically self-similar sets and a.s. self-similar sets and have got the Hausdorff dimension and exact Hausdorff measure function of any a.s. self-similar set in this paper. It is useful in the study of probability properties and fractal properties and structure of statistically recursive sets.     

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.     

Topological properties of self-similar fractals with one parameter

In this paper, we study two classes of planar self-similar fractals $T_{\epsilon }$ with a shifting parameter ε. The first one is a class of self-similar tiles by shifting x-coordinates of some digits. We give a detailed discussion on the disk-likeness (i.e., the property of being a topological disk) in terms of ε. We also prove that $T_{\epsilon }$ determines a quasi-periodic tiling if and only if ε is rational. The second one is a class of self-similar sets by shifting diagonal digits. We give a necessary and sufficient condition for $T_{\epsilon }$ to be connected.     

Lipschitz equivalence of self-similar sets and hyperbolic boundaries

Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009)  [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.     

Projections of random Cantor sets

Recently Dekking and Grimmett have used the theories of branching processes in a random environment and of superbranching processes to find the almostsure box-counting dimension of certain orthogonal projections of random Cantor sets. This note gives a rather shorter and more direct calculation, and also shows that the Hausdorff dimension is almost surely equal to the box-counting dimension. We restrict attention to one-dimensional projections of a plane set—there is no difficulty in extending the proof to higher-dimensional cases.     

Projections of self-similar sets with no separation condition

Two natural symplectic constructions, the Lagrangian suspension and Seidel’s quantum representation of the fundamental group of the group of Hamiltonian diffeomorphisms, Ham(M), with (M, ω) a monotone symplectic manifold, admit categorifications as actions of the fundamental groupoid Π(Ham(M)) on a cobordism category recently introduced in [BC14] and, respectively, on a monotone variant of the derived Fukaya category. We show that the functor constructed in [BC14] that maps the cobordism category to the derived Fukaya category is equivariant with respect to these actions.     

Enumeration problems for classes of self-similar graphs

We describe a general construction principle for a class of self-similar graphs. For various enumeration problems, we show that this construction leads to polynomial systems of recurrences and provide methods to solve these recurrences asymptotically. This is shown for different examples involving classical self-similar graphs such as the Sierpiński graphs. The enumeration problems we investigate include counting independent subsets, matchings and connected subsets.     

Topological measures,image transformations and self-similar sets

Summary This paper introduces a novel idea: the concept of an image transformation. We also introduce the closely related concept of a quasi-homomorphism, and study the properties of these mathematical objects, and give several examples. In particular we investigate iterated systems of image transformations, which we believe give a more realistic approach to the study of so called self-similar structures in nature than what is obtained by iterated function systems.     

On the variational principle for the topological pressure for certain non-compact sets

Let (X,d,T) be a dynamical system,where (X,d) is a compact metric space and T:X → X is a continuous map.We assume that the dynamical system satisfies g-almost product property and the uniform separation property.We compute the topological pressure of saturated sets under these two conditions.If the uniform separation property does not hold,we compute the topological pressure of the set of generic points.We give an application of these results to multifractal analysis and finally get a conditional variational principle.