Dimensions And Classification Of Moran-type Sets And Fractal Properties Of Gaussian Random Fields

Cut-out sets are one of the most important fractal sets. All compact subsets in R can be obtained in this way. In chapter 3, we focus on the cut-out set which is called Moran-type set. It could be obtained by gap sequence{ak}k≥1 and integer sequence {nk}k>1-Hausdorff measure is a fundamental tool to study fractal sets. As we known, even the Hausdorff dimension of a set E is α, the Hausdorff measure/Hα(E) may still be 0 or ∞. More general dimension function h and h-Hausdorff measure Hh(E) are needed to measure the size E. In chapter 3, we focus on cut-out set Ea which is called Moran-type cut-out set. We first give the upper and lower estimations of the h-Hausdorff and h-packing measures of Ea. Then we showed in the definition of general level sets the balls can be replaced by the intervals in Ea, as a result, we have that Ea can be written as the union of the general level sets. Additionally, we used three equivalent relations to characterise the equivalence of sets which are also very important topic in this area.The last two works are mainly to calculate the packing dimensions of the image and graph sets which induced by Gaussian random fields. More and more applications of Gaussian random fields have raised many interesting theoretical questions about Gaussian random fields in general. One of the major difficulties in studying the prob-abilistic, analytic or statistical properties of Gaussian random fields is the complexity of their dependence structures. As a result, many of the existing tools from theories on Brownian motion, Markov processes and martingales fail for Gaussian random fields; and one often has to use general principles for Gaussian processes or to develop new tools. Fractal dimensions such as Hausdorff dimension, box-counting dimension and packing dimension are useful for characterizing roughness and irregularities of stochas-tic processes and random fields. We refer to Taylor [56] and Xiao [62] for surveys on fractal properties of Markov processes, and to Adler [1], Kahane [31] and Xiao [65,66] for results on Gaussian random fields.1. Packing dimensions of the images of Gaussian random fieldsSimilar to the case of RN equipped with the Euclidean metric in Falconer and Howroyd [19], we introduce s-dimensional packing dimension profiles of E on the met-ric space (RN,ρ), denoted by DimsρE, where ρ is the pseudo-metric on RN defined by (1.0.2). For comparison purpose, we also extend the box dimension profiles and pack-ing dimension profiles with respect to a kernel of Howroyd [23] to (RN,ρ). Applying these more general forms of dimension profiles, in chapter 4 we determine the packing dimension of X(E) for an arbitrary Borel set E (?) RN by two kinds of packing dimen-sion profiles, which extend the known results of [64] and [30]. The uniform modulus of continuity of Gaussian random fields and potential theory play an important role in the proof.2. Packing dimensions of the graphs of fractional Brownian motionFractional Brownian motion is one of the most important Gaussian random fields. Xiao [61] determined the packing dimension of the image set of fractional Brownian motion, however there is few results on the graph set. In order to solve the problem, we make use of the metric τ on RN+d defined by (1.0.4). By extending the theory of the packing dimension profiles in [19], we determine the packing dimensions of the graphs of fractional Brownian motion in chapter 5. Our results solve the problem in some sense (see Problem 5 in Xiao [66]). Then we give the upper and lower estimations of the Cartesian product of the packing dimension profiles.