The Construction of Statistically Recursive Sets

We have constructed a class of random sets by statistical contraction operators in this paper.When the probability space is some special product space and the statistical contraction operators are affine or similar,these statistically recursive sets are investigated by many authors.It will be very convenient for us to study their distributions and dimensions and measures using our model in this paper.

The Classification of Statistically Recursive Sets

The main aim of this paper is to make a classification of random sets K m(ω) constructed in theorem 2.1 and theorem 2.1' in . We provide five criterions for the classification. Many kinds of random sets such as Hawkes constructed in , Graf constructed in and Mauldin constructed in are the special cases of K m(ω) constructed in ,and then these random sets belong to some model respectively according to our classification.

The Advances of the Research for Random Recursive Sets(Ⅰ)

We introduce the probability properties of random recursive sets systematically in this paper. The main contents include convergence, zero\|one law and support of distribution and self\|similarity.Hutchinson constructed a class of strictly self\|similar sets and got many important results on fractal properties.Graf investigated the fractal properties of a special statistically self\|similar set. We have investigated various self\|similar sets and their probability properties and fractal properties.\;

The Structure and the Sufficent and Necessary Conditions for Generalized Statistically Self-Similar Sets

We constructed a class of generalized statistically self-similar set.S and give the necessary and sufficent conditions to ensure a random recursive set being a generalized statistically self-similar set. The statistically self-similar sets defined by Hutchinson,Falconer,Graf are the special cases of ours.

HAUSDORFF DIMENSION OF GENERALIZED STATISTICALLY SELF-AFFINE FRACTALS

The authors consider generalized statistically self-affine recursive fractals K with random numbers of subsets on each level. They obtain the Hausdorff dimensions of K without considering whether the subsets on each level are non-overlapping or not. They also give some examples to show that many important sets are the special cases of their models.

Bounded Recursively Enumerable Sets and Degrees

A new reducibility between the recursive sets is defined,which is appropriate to be used in the study of the polynomial reducibility and the NP-problem.

Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

We prove that there are non-recursive r.e.sets A and C with A<T C such that for every set F(?)T A,C∩F≡w(?).