分形测度的点态维数、L~q-谱及不正则集
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摘要
重分形分析是分形几何和动力系统的一个重要分支。重分形测度及重分形分析的概念首先由一些物理学家[39]提出。Barreira, Pesin和Schmeling[5]提出了如下一般的重分形框架。设X是一个恰当的集合,Y∈X,考虑函数g：Y→[—∞,+∞].显然,g的水平集是不交的。从而,X有以下重分形分解：另外,设G是定义在X的所有子集上满足单调性的集函数。定义函数F：[—∞,+∞]→R为：我们称F是由函数对(g,G)生成的重分形谱。有很多很自然的方式选取g和G,参见文献[5].式(0-1)中的集合X\Y是使得函数g没有定义的点组成的集合,我们称之为不正则集。
在以上一般的重分形框架下,我们需要面临如下两个问题：(1)函数g的定义域,即函数g的存在性及不正则集的刻画；(2)各种重分形谱F的计算或估计。本文主要围绕这两个问题,对某些分形测度展开研究。
本文分为三部分。
1.一类满足开集条件的Moran测度的点态维数
测度是研究分形集的基本工具。测度的局部性质的研究在分形几何中显得尤为重要。本文第三章我们利用Moran网填充测度,建立了关于填充维数的Billingsley定理。利用测度的维数和点态维数之间的关系以及Billingsley定理,得到了一类满足开集条件的Moran测度的上(下)点态维数公式(在几乎处处的意义下)。作为应用,我们还得到了这类Moran测度的Hausdorff维数和填充维数。
2.无分离条件自相似测度的Lq-谱
Lq-谱在测度的重分形分析中扮演重要角色。在第四章我们证明了当q≤1时一般自相似测度的上填充Renyi维数不会超过Lq-谱这一结果,然后结合自相似测度的上覆盖Renyi维数和Lq-谱之间的关系,得到当q≤1时自相似测度的Lq-谱的非平凡估计,回答了Olsen [66]提出的一个问题。作为一个应用,我们得到无分离条件的自相似测度重分形谱的一个非平凡上界估计,并讨论了两个不满足开集条件的自相似测度的例子：(2,3)-Bernoulli卷积和λ-Cantor测度。
3.两类不正则集的刻画
一般而言,不正则集从(不变)测度的角度讲是零测集。正因为如此,不正则集在很长一段时间内被认为不重要而被忽视。但是,近来越来越多的文献表明不正则集从维数角度而言是很“大”的,且具有丰富的分形结构,见文献[4,6,7,20,36,51,55,64,65,81]及其引文。
本文第五章我们讨论了开集条件下自相似测度精细不正则集的维数。设μ是支撑在自相似集K上满足开集条件的自相似测度。对χ∈K,令A(D(χ))表示当r↘0时函数Dr(x):=(logμ(B(x,r)))/(logr)的聚点集。我们证明了对任意χ∈K,集合A(D(χ))要么是一单点集要么是一个闭区间。对任意闭区间I∈R,我们利用Vitali覆盖引理,加细型计盒原理以及构造Moran子集的技巧,计算了使得A(D(χ))=I的点χ组成集合(自相似测度精细不正则集)的Hausdorff维数和填充维数。我们的结果解决了Olsen和Winter[64]提出的一个猜想,并且包含了Arbeiter和Patzschke[1]的经典结果。
本文第六章我们讨论具有specification性质的动力系统的不正则集的拓扑熵。设(X,f)是一拓扑动力系统。类似于自相似测度精细不正则集的定义,我们定义了Birkhoff平均的精细不正则集。在(X,f)满足specification性质的假设下,我们利用熵分布原理及构造动力Moran子集的技巧,计算了Birkhoff平均的精细不正则集的拓扑熵。我们的结果包含了Takens和Verbitskiy[79]的经典结果,并且再次显示了不正则集从维数的角度看可以很“大”、可以具有丰富的结构。作为应用,我们给出了Birkhoff平均的不正则点集具有满拓扑熵的一个简洁证明。
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 measures
Measure 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 sets
Generally 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.
在以上一般的重分形框架下,我们需要面临如下两个问题：(1)函数g的定义域,即函数g的存在性及不正则集的刻画；(2)各种重分形谱F的计算或估计。本文主要围绕这两个问题,对某些分形测度展开研究。
本文分为三部分。
1.一类满足开集条件的Moran测度的点态维数
测度是研究分形集的基本工具。测度的局部性质的研究在分形几何中显得尤为重要。本文第三章我们利用Moran网填充测度,建立了关于填充维数的Billingsley定理。利用测度的维数和点态维数之间的关系以及Billingsley定理,得到了一类满足开集条件的Moran测度的上(下)点态维数公式(在几乎处处的意义下)。作为应用,我们还得到了这类Moran测度的Hausdorff维数和填充维数。
2.无分离条件自相似测度的Lq-谱
Lq-谱在测度的重分形分析中扮演重要角色。在第四章我们证明了当q≤1时一般自相似测度的上填充Renyi维数不会超过Lq-谱这一结果,然后结合自相似测度的上覆盖Renyi维数和Lq-谱之间的关系,得到当q≤1时自相似测度的Lq-谱的非平凡估计,回答了Olsen [66]提出的一个问题。作为一个应用,我们得到无分离条件的自相似测度重分形谱的一个非平凡上界估计,并讨论了两个不满足开集条件的自相似测度的例子：(2,3)-Bernoulli卷积和λ-Cantor测度。
3.两类不正则集的刻画
一般而言,不正则集从(不变)测度的角度讲是零测集。正因为如此,不正则集在很长一段时间内被认为不重要而被忽视。但是,近来越来越多的文献表明不正则集从维数角度而言是很“大”的,且具有丰富的分形结构,见文献[4,6,7,20,36,51,55,64,65,81]及其引文。
本文第五章我们讨论了开集条件下自相似测度精细不正则集的维数。设μ是支撑在自相似集K上满足开集条件的自相似测度。对χ∈K,令A(D(χ))表示当r↘0时函数Dr(x):=(logμ(B(x,r)))/(logr)的聚点集。我们证明了对任意χ∈K,集合A(D(χ))要么是一单点集要么是一个闭区间。对任意闭区间I∈R,我们利用Vitali覆盖引理,加细型计盒原理以及构造Moran子集的技巧,计算了使得A(D(χ))=I的点χ组成集合(自相似测度精细不正则集)的Hausdorff维数和填充维数。我们的结果解决了Olsen和Winter[64]提出的一个猜想,并且包含了Arbeiter和Patzschke[1]的经典结果。
本文第六章我们讨论具有specification性质的动力系统的不正则集的拓扑熵。设(X,f)是一拓扑动力系统。类似于自相似测度精细不正则集的定义,我们定义了Birkhoff平均的精细不正则集。在(X,f)满足specification性质的假设下,我们利用熵分布原理及构造动力Moran子集的技巧,计算了Birkhoff平均的精细不正则集的拓扑熵。我们的结果包含了Takens和Verbitskiy[79]的经典结果,并且再次显示了不正则集从维数的角度看可以很“大”、可以具有丰富的结构。作为应用,我们给出了Birkhoff平均的不正则点集具有满拓扑熵的一个简洁证明。
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 measures
Measure 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 sets
Generally 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.
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