Devil's-Staircase

Devil's-Staircase - The Devils Staircase Recall the usual...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: The Devils Staircase Recall the usual construction of the Cantor set: C = [0 , 1] , C 1 = [0 , 1 3 ] [ 2 3 , 1] , and in general C n is a disjoint union of 2 n closed intervals, each of length 3- n , constructed from C n- 1 by deleting the open-middle-third of each of the 2 n- 1 intervals constituting C n- 1 . Then the total length of the intervals in C n is ( 2 3 ) n . The Cantor set C is the intersection C = T n C n . Set g n = ( 3 2 ) n 1 C n ; that is, g n ( x ) = ( ( 3 2 ) n , x C n , x / C n . The function g n is discontinuous only at the 2 n +1 points at the boundaries of the intervals making up C n . This is a finite set, and so g n is Riemann integrable. So we may define f n : [0 , 1] R as follows: f n ( x ) = Z x g n ( t ) dt. From the Fundamental Theorem of Calculus, we know that the functions f n are Lipschitz continuous. Note also that f n (0) = 0 , while f n (1) = R 1 g n ( t ) dt . This integral can be calculated as Z 1 g n ( t ) dt = ( 3 2 ) n R 1 1 C n ( t ) dt = ( 3 2 ) n length ( C n ) = 1 . So f n is a continuous function with f n (0) = 0 and f n (1) = 1 for each n . Notice that f n is constant on [0 , 1]- C n , and is linear with slope ( 2 3 ) n on the intervals making up C n . So f n is monotone increasing. Here is the graph ofis monotone increasing....
View Full Document

This note was uploaded on 12/07/2011 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.

Page1 / 3

Devil's-Staircase - The Devils Staircase Recall the usual...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online