| 4. (a) If the generator is unifractal, then |dYi| = (dti)H, with the same H for i = 1, 2, and 3. This gives dti = |dYi|1/H. |
| (b) To find the Trading time generator, first solve |
| |dY1|D + |dY2|D + |dY3|D = 1. |
| Using this D, the Trading time generators are dTi = |dYi|D. |
| Now we know 1 = dt1 + dt2 + dt3 = |dY1|1/H + |dY2|1/H + |dY3|1/H |
| Also we know |dY1|D + |dY2|D + |dY3|D = 1. |
| Finally, we know this equation has a unique solution (this is just the proof of the uniqueness of the solution of the Moran equation), so D = 1/H. |
| Because dti = |dYi|1/H and dTi = |dYi|D, we see dti = dTi. |
| (c) This shows the Trading time theorem leaves unifractal generators unaltered. |
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