tailieunhanh - Báo cáo "REMARKS ON LOCAL DIMENSION OF FRACTAL MEASURE ASSOCIATED WITH THE (0, 1, 9) - PROBLEM "

Let X be random variable taking values 0, 1, a with equal probability 1/3 and let X1 , X2 , . be a sequence of independent identically distributed () random variables with the same distribution as X. Let µ be the probability measure induced by ∞ S = i=1 3−i Xi . Let α(s) (resp. α(s), α(s)) denote the local dimension (resp. lower, upper local dimension) of s ∈ supp µ. Put | VNU. JOURNAL OF SCIENCE Mathematics - Physics. N03 - 2005 REMARKS ON LOCAL DIMENSION OF FRACTAL MEASURE ASSOCIATED WITH THE 0 1 9 - PROBLEM Truong Thi Thuy Duong Le Xuan Son Pham Quang Trinh Department of Mathematics Vinh University Vu Hong Thanh Pedagogical College of Nghe An Abstract. Let X be random variable taking values 0 1 a with equal probability 1 3 and let X1 X2 . be a sequence of independent identically distributed random variables with the same distribution as X. Let p be the probability measure induced by S y 1 3-iXị. Let a s resp. a s a s denote the local dimension resp. lower upper local dimension of s E supp p. Put a sup a s s E supp p a inf a s s E supp p E a a s a for some s E supp p . When a 0 mod 3 the probability measure p is singular and it is conjectured that for a 3k for any k E N the local dimension is still the same as the case k 1 2. It means E 1 j 11 for a b depend on k. Our result shows that for k 3 a 9 a 1 b log 3 a 2 3 and E 2 1 . 1. Introduction By a Probabilistic system we mean a sequence X1 X2 . of random variables with the same distribution as X where X is a random variable taking values ai a2 . am with probability P1 P2 . pm respectively. Let S y plXị for 0 p 1 then the probability measure p induced by S . p A Prob w S w E A is called the Fractal measure associated with the probabilistic system. It is specified two interesting cases. The first case is when m 2 p1 P2 1 2 and a1 0 a2 1. In this case the fractal measure p is known as Infinite Bernoulli Convolutions . This measure has been studied for over sixty years but is still only partial understood today. Typeset by AmS-Ti X 33 34 Truong Thi Thuy Duong Le Xuan Son Pham Quang Trinh Vu Hong Thanh The second case of interest is when m 3 p P1 P2 P3 1 3 and a1 0 a2 1 a3 a. Some authors have found a a and E for some concrete values of a for which p is singular. But for each of them the way to find a a and E is quite different. It is conjectured that the way to .