Assignment: Page 131: Nos. 1-4, 6-13, 15, 16, 21, 22, 24.
For each of the following sets , decide whether or not the set is dense in [0,1].
We will do this by using a bisection technique on the interval [w,2w].
Begin by letting x0 = 2w = 1/2^n-1 and form the sequence
xk = ( w + x(k-1) )/ 2 for k>0
=> x1=3/2^n+1, x2=5/2^n+2, x3=9/2^n+3 and in general
xk=(2^k + 1)/2^n+k {note xk is in S1 for each k>0}
Claim, xk -> w as k -> infinity.
To see this, consider the function Fw(x)= (w+x)/2 where w is any real
number. Notice that Fw is a linear map with attracting fixed pt w and basin
of attraction all of [0,1].
In Summary, let w = 1/2^n be a point not in S1, x0=2w and Fw(x)=(w+x)/2. Then
Fw^k(x) -> w as k -> infinity, and Fw^k(x) is in S1 for all k>0. This
proves that S1 is dense in [0,1].
Note that S_2 contains all multilples of 1/2, all multiples of 1/4, etc. In fact, x is in S_2 if and only if x has a terminating binary expansion. Assuming this to be true for the moment, we may easily show that S_2 is dense in [0,1] as follows: let w = 0.b1b2b3... be an arbritary point in [0,1]. If w terminates, then we are done, so suppose it dies not. Then the sequence,
Recall that the ternary expansion of x in S_4 has no 1's. Any point which does have a 1 in its ternary expansion is in S_4's complement. Moreover, any finite string can be prepended to such a point, and the result is still in S_4's complement. Thus the complement of S_4 is dense in [0,1] since there are (uncountably many) pints in S_4's complement arbitrarily close to any point x in the Cantor set.
No-in fact, the orbit of (01 001 0001 00001 ...) stays away from M_11={sequences with s_0=1,s_1=1} altogether. and there's really nothing special about the systematically increasing number of 0's in this string. No element of
Consider the orbit of
We will show that the doubling map D is semi-conjugate to the tent map T via T itself! That is, we will show that
We eill show that
Our goal si to find a linear map W:[-2,2] -->[0,1] such that