| 11. (a) The probability of each address is the product of the probabilities of
applying the transformations corresponding to that address: |
| address | probability |
| 111 | (2/5)3 = .064 |
| 112 | (2/5)2(3/5) = .096 |
| 121 | (2/5)2(3/5) = .096 |
| 122 | (2/5)(3/5)2 = .144 |
| 211 | (2/5)2(3/5) = .096 |
| 212 | (2/5)(3/5)2 = .144 |
| 221 | (2/5)(3/5)2 = .144 |
| 222 | (3/5)3 = .216 |
|
| (b) The Hölder exponent of an address is Log(prob)/Log(interval length).
For each length 3 address, the interval length is 1/8. |
| address | Hölder exponent |
| 111 | Log((2/5)3)/Log(1/8) = 1.322 |
| 112 | Log((2/5)2(3/5))/Log(1/8) = 1.127 |
| 121 | Log((2/5)2(3/5))/Log(1/8) = 1.127 |
| 122 | Log((2/5)(3/5)2)/Log(1/8) = .932 |
| 211 | Log((2/5)2(3/5))/Log(1/8) = 1.127 |
| 212 | Log((2/5)(3/5)2)/Log(1/8) = .932 |
| 221 | Log((2/5)(3/5)2)/Log(1/8) = .932 |
| 222 | Log((3/5)3)/Log(1/8) = .737 |
|
| (c) |
| Hölder exponent | number of intervals |
| 1.322 | 1 |
| 1.127 | 3 |
| .932 | 3 |
| .737 | 1 |
|
| (d) Take Log of both sides |
| Log(Num(α)) = Log((1/8)-f(α)) = -f(α)Log(1/8) |
| Solving for f(α) gives |
| f(α) = -Log(Num(α))/Log(1/8) |
| (e) |
| α | f(α) |
| 1.322 | -Log(1)/Log(1/8) = 0 |
| 1.127 | -Log(3)/Log(1/8) = .528 |
| .932 | -Log(3)/Log(1/8) = .528 |
| .737 | -Log(1)/Log(1/8) = 0 |
|