tailieunhanh - Báo cáo " The extreme value of local dimension of convolution of the cantor measure"

Let µ be the m−fold convolution of the standard Cantor measure and αm be the lower extreme value of the local dimension of the measure µ. The values of αm for m = 2, 3, 4 were showed in [4] and [5]. In this paper, we show that | VNU Journal of Science Mathematics - Physics 25 2009 57-68 The extreme value of local dimension of convolution of the cantor measure Vu Thi Hong Thanh1 Nguyen Ngoc Quynh2 Le Xuan Son1 1 Department of Mathematics Vinh Universily 2 Department of Fundamental Science Vietnam academy of Traditional medicine Received 5 March 2009 Abstract. Let p be the m fold convolution of the standard Cantor measure and am be the lower extreme value of the local dimension of the measure p. The values of am for m 2 3 4 were showed in 4 and 5 . In this paper we show that 1 4 arccos797745 I Ml a 1og 3W r 3 3 . This values was estimated by P Shmerkin in 5 but it has not been proved. Key words Local dimension probability measure standard Cantor measure. 2000 AMS Mathematics Subject Classification Primary 28A80 Secondary 42B10. 1. Introduction m Let Sj m 1 be contractive similitudes on Rd and pj m 0 Pj 1 52 Pj 1 be a set of j 1 probability weights. Then there exists a unique probability measure p satisfying m p A Pj p S-1 A j 1 for all Borel measurable sets A see 1 . We call p a self-similar measure and Sj j 1 a system iterated functions. When S1 . Sm are similarities with equal contraction ratio p E 0 1 on R . Sj x p x bj bj E R for j 1 . m the self-similar measure p can be seen as follows Let Xo X1 . be a sequence of independent identically distributed random variables each taking real values b1 . bm with probability P1 . Pm respectively. We define a random variable S 52 piXi then the probability i 1 measure pp induced by S pp A P w S w E A is called a fractal measure and pp p see 2 . Let V be the standard Cantor measure then V can be considered to be generated by the two maps Si x 3x 2i i 0 1 with weight 1 on each Si. Then the attractor of this system Corresponding author. E-mail vu_hong_thanh@ 57 58 . Thanh et al. VNU Journal of Science Mathematics - Physics 25 2009 57-68 iterated functions is the standard Cantor set C . C S0 C u S1 C . Let p V . V be .