Exact Hausdorff Dimension in Random Recursive Constructions - download pdf or read online

By Siegfried Graf

ISBN-10: 0821824449

ISBN-13: 9780821824443

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1 this implies r = 0. //»)) < J p_a. 38) H c > —=-± 1 n k! 3*. 1 3 ! ''n* T aj R i «• i = l - C k n x A' R i=l i j,//3 n Since the function ( y ^ . . ,j /k) S. D. MAULDIN, S. 39) E [lf " 3 (Tia/(1-(1//5)))(1~(1//5))ji] T^^i] = E [ j f i ill "^ 1: r(x) < x x - ( 1 / 2 ) e x e1/12x y E T and _, . v> xx-(l/2) //—2TT K— . {1-(1//5)):ii TT 2 1=1 (l-fi/zmk y~2F") k ( 1 <1","k EXACT DIMENSION IN RANDOM CONSTRUCTIONS 3 n JJ ( j , (Jl//») + ( l / 2 ) * + 1) * (k + 1)(k//») < [en+1(k + i ) r n ill!

98) V t > tQ : (1/2 )E(( } P=l % ) * JT ^ ( T . 99) (1/2)E[( ^ T ) a-v J Tn P=l 1 (¥ . 11. 13. 101) n ? ~ t T )t E(( I P=l JT p i 1 { T . 102) Proof. k->«> s, ( 1 / 0 ) K > 0. 14. ,T ) has a derivative f > 0 with respect to Lebesgue measure on A « 36 S. D. MAULDIN, S. WILLIAMS {(Yjf-'-fY ) € (Ofl)n| \ y. < 1}. Moreover, suppose that there i=l exists a point x = (x.. , . . 104) x. = 1 and x. 105) there is some c>0 and a neighborhood U of x with f(y) > c for all y € U n A 1 n with y. < x. ,n. Then Urn k y s k (l/d) > 0.

N} \ B ) . It is easy to check that r is an antichain. c f. 44) Hence there exists a maximal antichain r with ? 40) there exists a k € ft with [log k] > k Y ^(w,(logllog O € B (W) k e (w) G | ) 1 / / 5 * *• and S. D. MAULDIN, S. WILLIAMS 50 For every a € B we have o|k € B (w) . 46) ^(K(«)) < t 1//5 X(w) < and proves the theorem. 4. A LOWER BOUND FOR THE HAUSDORFF DIMENSION Throughout this section we assume that diara J = 1 and P(z n T? * 1) > 0. Rm. Lemma 4,1. , an increasing continuous function defined for small t > 0 and vanishing at 0).

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Exact Hausdorff Dimension in Random Recursive Constructions by Siegfried Graf


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