THE HAUSDORFF DIMENSION AND EXACT HAUSDORFF MEASURE OF RANDOM RECURSIVE SETS WITH OVERLAPPING

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.


Introduction.
As it is known, the separation conditions, such as the strong separation condition, the open set condition (OSC), and the strong open set condition, must be taken into consideration when computing the Hausdorff dimensions of the random recursive sets. In deterministic cases, Schief [13] proved that the strong open set condition and the open set condition are both equivalent to ∞ > Ᏼ α (K) > 0, where K is the strictly self-similar set (cf. Hutchinson [9]) in R d , α is the similarity dimension of K, and Ᏼ α denotes the Hausdorff measure of this dimension. But in random cases, we do not have such good results, many authors, such as Cawley and Mauldin [2], Falconer [3], Graf [6], Mauldin and Williams [12], Arbeiter and Patzschke [1], and Hu [7,8], have discussed the fractal properties of the random recursive set K(ω), and the most general result may be: if the open set condition is satisfied in the random recursive process of i.i.d. contraction similitudes, then dim K(ω) = α with probability one, where α is the unique solution of the equation and E is the expectation operator and r i is the Lipschitz coefficients of the similitudes. Sometimes the open set condition in the construction of recursive sets is complex and difficult to verify. In this paper, we try to find another criterion to calculate the fractal dimensions of some random recursive sets, we give a definition of the finite intersection property (FIP) which allows appropriate overlapping in the same level. This condition is rather easy to verify, especially in the generalized Moran sets and Mauldin-Williams (M-W) models [12] (in fact, the open set condition is equivalent to the nonoverlapping in the same level in the recursive process of M-W models). We prove that if the recursive process satisfies the OSC, then it satisfies the FIP, and we give examples which satisfy FIP but do not satisfy OSC; we also prove the following theorem.
Throughout the paper, we suppose that r = min σ ∈Cn,n≥1 essinf Lip(S σ ) > 0 and max σ ∈Cn,n≥1 ess sup Lip(S σ ) < 1, and E is a fixed nonempty compact subset of R d . For a set J ⊂ E, let J 0 denote the set of interior points of J.
where ᏸ is the Lebesgue measure on R d and V d is the Lebesgue measure of the unit ball in R d . So, and Lemma 2.4 is proved.

Proof. By the definition of the metric ρ * , the cylinder [σ ] is both open and closed, since
for every σ ∈ D. Let Ꮾ be the collection of all cylinders [σ ], σ ∈ D, and let Ꮽ be the collection of a finite union of disjoint cylinders, where by convention the empty union is taken to be the empty set ∅. Then Ꮽ is an algebra. Define a random set function ν ω by ν(∅) = 0, ν([σ ]) = r α (|σ |,σ ) Y σ , for σ ∈ D, then by (3.8) we have that so for almost every ω, the set function ν is well defined. By the compactness of C, it can be easily seen that if A n ∈ Ꮽ decreases to ∅, then A n = ∅ for n large enough, so that ν(A n ) decreases to ∅; this shows that ν is a measure on Ꮽ. In a natural way, we can extend ν to a Borel measure on σ (Ꮾ) (cf. [11]). In fact, (3.14) Proof. Since {J (n,σ ) , σ ∈ C n } is a covering of K for almost all ω ∈ Ω, and r (n,σ ) → 0, we have Suppose that N i=1 r α i = 1 a.e., then Y = 1 a.e. by its definition. If Ᏼ α (K) < 1/l 1 a.e., then there would be a collection Ᏹ of sets each with diameter less than r and covering K such that E∈Ᏹ diam(E) α < 1/l 1 a.e. But taking = 0 in (3.15), (3.17) This would lead to a contradiction. So this proposition holds.

Examples.
First, we give an example which satisfies the FIP but not the OSC. (I) The first two steps: (4.1) Thus we have four basic intervals.
(II) The second two steps: as to the interval [0, 1/9], we repeat the same construction technique as in (I). As to the rest of the three basic intervals, we can easily find maps with ratios {r σ * 1 ,r σ * 2 } = {1/3, 1/27} for σ ∈ C 3 and C 4 , such that the subsets of all these three basic intervals are disjoint. And so on.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos). We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009