| 14. (a) The highest probabilities occur on length 2 addresses 11, 12, 21, 22, and on
length 3 addresses 111, 112, 121, 122, 211, 212, 221, 222. The lowest probabilities occur on
length 2 addresses 33, 34, 43, 44, and on length 3 addresses 333, 334, 343, 344, 433, 434,
443, 444. The highest probabilities are 0.42 and 0.43.
The lowest probabilities are 0.12 and 0.13. |
| (b) The corresponding α values are |
| α(11) = ... = α(22) | = Log(0.42)/Log(0.52) | = Log(0.4)/Log(0.5) ≈ 1.322 |
| α(33) = ... = α(44) | = Log(0.12)/Log(0.52) | = Log(0.1)/Log(0.5) ≈ 3.322 |
| α(111) = ... = α(222) | = Log(0.43)/Log(0.53) | = Log(0.4)/Log(0.5) ≈ 1.322 |
| α(333) = ... = α(444) | = Log(0.13)/Log(0.53) | = Log(0.1)/Log(0.5) ≈ 3.322 |
|
| (c) For each n, the minimum α value is Log(0.4n)/Log(0.5n) ≈ 1.322. |
| For each n, the maximum α value is Log(0.1n)/Log(0.5n) ≈ 3.322. |
| (d) For each n, the minimum α value occurs on the region whose addresses contain only 1 and 2. In the
n → ∞ limit, this is the set determined by the IFS T1 and T2, that is, a straight
line segment, of dimension 1. |
| Similarly, in the → ∞ limit, the maximum α value occurs on the set determined by the IFS
T3 and T4, another straight line segment, of dimension 1. |