tailieunhanh - Báo cáo " SINGULARITY OF FRACTAL MEASURE ASSOCIATED WITH THE (0, 1, 7) - PROBLEM "

Let µ be the probability measure induced by S = n=1 3 X1 , X2 , . is a sequence of independent, identically distributed () random variables each taking values 0, 1, a with equal probability 1/3. Let α(s) ((s), α(s)) denote the local dimension (resp. lower, upper local dimension) of s ∈ supp µ, and let | VNU. JOURNAL OF SCIENCE Mathematics - Physics. No2 - 2005 SINGULARITY OF FRACTAL MEASURE ASSOCIATED WITH THE 0 1 7 - PROBLEM Truong Thi Thuy Duong Vinh University Nghe An Vu Hong Thanh Pedagogical College of Nghe An Abstract. Let g be the probability measure induced by S 2m i 3-nXn where X1 X2 . is a sequence of independent identically distributed random variables each taking values 0 1 a with equal probability 1 3. Let a s s a s denote the local dimension resp. lower upper local dimension of s E supp g and let a sup a s s E supp g a inf a s s E supp g E a a s a for some s E supp g . In the case a 3k 1 for k 1 E 1 log 1 0g53 log2 1 see 10 . It is conjectured that in the general case for a 3k 1 k E N the local dimension is of the form as the case k 1 . Ei 1 b log 3 1 for a b depends on k . In fact our result shows that for k 2 a 7 we have a 1 a 1 l 3k 3 and E 1 log x 33 1 . 1. Introduction Let X be random variable taking values a1 a2 . am with probability P1 P2 . pm respectively and let X1 X2 . be a sequence of independent random variables with the same distribution as X. Let S sn i pnXn for 0 p 1 and let g be the probability measure induced by S . g A Prob w S w E A . It is known that the measure is either purely singular or absolutely continuous. An intriguing case when m 3 p P1 P2 P3 1 3 and a1 0 a2 1 a3 a. According to the pure theorem of Lagarias and Wang in 7 if a 0 mod 3 or a 1 mod 3 then g is purely singular. Let us recall that for s E supp g the local dimension a s of g at s is defined by a s lim logg Bh s 1 h . log h Typeset by ẠmS-T .ịX 7 8 Truong Thi Thuy Duong Vu Hong Thanh provided that the limit exists where Bh s denotes the ball centered at s with radius h. If the limit 1 does not exist we define the upper and lower local dimension denoted a s and a s by taking the upper and lower limits respectively. Denote a sup a s s E supp y a inf a s s E supp y and let E a a s a for some s E supp y be the attainable values of a s . .