三部分cantor的-华东师范大学.pdfVIP

3 ( ) No. 3 2011 5 Journal of East China Normal University (Natural Science) May 2011 : 1000-5641(2011)03-0054-05 Cantor , (, 200241) : Cβ β ∈ (0, 1/3) Cantor . : β ∈ (0, 1/3), (0, 0) λ Lλ (λ ∈ (0, 1)), Cβ × Cβ . . : Cantor ; ; : O174.1 : A DOI: 10.3969/j.issn.1000-5641.2011.03.008 Geometrical property of the three-part homogenous Cantor-set LI Xue-feng, JIANG Kan (Department of Mathematics, East China Normal University, Shanghai 200241, China) Abstract: Let Cβ be the three-part homogenous cantor-set with ratio β ∈ (0, 1/3). This paper studied a visibility problem: given β ∈ (0, 1/3), is there a ray Lλ with slope λ ∈ (0, 1) which goes through the origin (0, 0) such that the intersection of Lλ and set (Cβ × Cβ ) only contains the origin? Based on the method of renormalization, some results were obtained. Key words: three-part homogenous cantor-set; affine map; renormalization 0 Cantor , [1-3]. 1998, Kraft [1] Cantor . E Cantor , [0, 1] {f (x) = βx, f (x) = βx + 1 − β } 1 2 , β ∈ (0, 1/2). : β ∈ (0, 1/2), λ ∈ (0, 1) E ∩ λE = {0}? : β ∈ (0, 1/2), Lλ : y = λx(λ ∈ (0, 1), x 0) (E × E) ∩ Lλ = {(0, 0)}? Kraft : (i) β ∈ (0, 3−√5 ), λ ∈ (0, 1) E ∩ λE = {0}. 2 (ii) β ∈ [3−√5 , 1 ), λ ∈ (0, 1) E ∩ λE = {0}(E λE 2 2 0). : 2010-10 : ; (11ZZ410) : , , . E-mail: wjss3600@126.com. 3 , : Cantor