Hausdorff dimension and measure of basin boundaries

✍ Scribed by Karen M Brucks

• Publisher: Elsevier Science
• Year: 1989
• Tongue: English
• Weight: 976 KB
• Volume: 78
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(89)90032-7

✦ Synopsis

To begin, we consider the following example. Fix a value e E (0, l/2). Let g be the piecewise linear map shown in Fig. 1, where A is chosen so that l/2 is a super stable periodic point of period 3,1> l/2, and g(n) < l/2 (it is known [2, 3, 25, 261 that, given an e, such a II exists).

Thus, under iteration of the map g, l/2 is first sent to the right of l/2, then to the left of l/2 and finally back to l/2, i.e., the kneading sequence for the map g is RLC. Let B' be the basin of attraction for this stable three period, i.e., the orbit of any point in g converges to the orbit of this stable three period. Let % be the boundary of 9?. In this paper we show the following:

(1) the Hausdorff dimension of G9 is WWW4~ where cp denotes the golden mean (1+ sqrt(5))/2, and

(2) the measure of %' in its dimension is finite and positive.