Pointwise Dimensions, L~q-spectra And Irregular Sets Of Fractal Measures

The multifractal analysis is an important branch of fractal geometry and dynamical systems. The concepts of multifractal measures and multifractal analysis were firstly introduced by some physicist [39]. Barreira, Pesin and Schmeling proposed a general concept of multifractal analysis [5]. Let X be a set, Y C X and let g:Y→[-∞,+∞] be a function. The level sets of g are disjoint and produce a multifractal decomposition of X, i.e., Let now G be a set function, i.e., a real function that is defined on subsets of X. Assume that G(Z1)≤G(Z2) if Z1(?)Z2. We define the function F=:[-∞,+∞]→R by We call F the multifractal spectrum specified by the pair of functions (g,G). There are many natural ways to choose g and G, see [5]. The set X\Y in equation (0-1) insists of those points at which g has no sense and we call it irregular set.On the above general setting, we face the following two problems:(1) the domain of the function g, i.e., the existence of the function g and the characterization of the irregular set;(2) the calculation or estimate of the multifractal spectrum F. This dissertation will study some fractal measures according to the above two problems.This dissertation consists of three parts.1. The pointwise dimension of a class of Moran measuresMeasure is a fundamental tool for study fractal sets. The study of local property of fractal measures plays an important role in fractal geometry. In chapter3we will discuss the pointwise dimension of a class of Moran measure. We establish the Billingsley theorem with respect to packing dimension with the help of the Moran net packing measure, then obtain the upper and lower pointwise dimensions of the Moran measures (in the sense of almost everywhere) by the relationships between pointwise dimension and dimension of measure and the Billingsley's theorem. As an application of our result, we obtain the Hausdorff and packing dimensions of the Moran measures. 2. Lq-spectra of self-similar measures without any separation condition.Lq-spectra play an important role in multifractal analysis. In chapter4we will discuss the Lq-spectra of self-similar measures without any separation conditions. We prove that the upper packing Renyi dimension less than the Lq-spectrum, then we obtain the nontrivial estimate of the Lq-spectrum of any self-similar measure for q≤1by the relationships between upper cover Renyi dimension and Lq-spectrum of self-similar measure. As an application we obtain non-trivial upper bounds for the multifractal spectra of arbitrary self-similar measure without any separation conditions, and discuss two self-similar measures which do not satisfy the open set condition:(2,3)-Bernoulli convolution and λ-Cantor measure.3. Two classes of irregular setsGenerally speaking, the irregular set is not detectable from the point of view of an invariant measure. Just because of this they are regarded as unimportant in a long period. However, it is an increasingly well known phenomenon that the irregular set can be large from the point of view of dimension theory [4,6,7,20,36,51,55,64,65,81].In chapter5we will discuss the refined irregular sets of self-similar measures with the open set condition. Let μ be the self-similar measure supported on the self-similar set K with the open set condition. For x∈K, let A(D(x)) denote the set of accumulation points of Dr(x):=(logμ(B(x,r)))/(logr) as r↘0. We show that the set A(D(x)) is either a singleton or a closed subinterval of R for any x∈K. For any closed subinterval I(?)R, we determine the Hausdorff dimension and packing dimension of the set of points x for which the set A(D(x)) equals I, using the Vitali cover lemma, refined box-counting principle and the technique of constructing Moran subset. Our main result solves the conjecture posed by Olsen and Winter [64] positively and generalizes the classical result of Arbeiter and Patzschke [1].In chapter6we discuss the refined irregular sets of systems satisfying the specifi-cation. Let (X,f) be a topological dynamical system. Similar to the definition of the refined irregular set of self-similar measure, we define the refined irregular set of the Birkhoff average. Under the hypothesis that f satisfies the specification property, us-ing the entropy distribution principle and he technique of constructing dynamical Moran subset, we determine the topological entropy of the refined irregular set of the Birkhoff average. Our result generalizes the classical result of Takens and Verbitskiy [79]. As an application, we present another concise proof of the fact that the irregular set has full topological entropy if f satisfies the specification property.