| A simple way to construct multifractals is to use an IFS with transformations
{T1, ..., TN}, contraction ratios r1, ..., rN,
and probabilities p1, ..., pN. |
| We will show how to construct the f(alpha) curve from this information, through an
auxiliary function tau(q) defined by the equation. |
| (p1q)(r1tau(q)) + ...
+ (pNq)(rNtau(q)) = 1 |
| The similarity to the Moran equation |
| r1d + r2d + ... + rNd = 1 |
| is apparent. |
| A reason for this approach is that for large positive q the larger pi dominate;
for large negative q the the smaller pi dominate. |
| While only two of the steps are subtle, we break the development into several pieces. |
| First, we show that each q determines a unique value of tau(q). |
| Next, tau(q) -> infinity as q -> -infinity, and
tau(q) -> -infinity as q -> infinity. |
| Tau is a decreasing function of q and is concave up. |
| Corresponding to each q, say alpha is the negative of the slope of the tangent to the graph of tau(q).
This tangent line intersects the y-axis at a value called f(alpha). |
| The range of alpha values is min{log(pi)/log(ri)}
= alphamin <= alpha <= alphamax = max{log(pi)/log(ri)}. |
| The graph of tau(q) has oblique asymptotes, with slopes -alphamax
as q -> -infinity and -alphamin as q -> infinity. |
| If all the log(pi)/log(ri) are distinct, then f(alphamin) =
f(alphamax) = 0. |
| If not all the log(pi)/log(ri) are equal, then the graph of f(alpha) is
concave down. |
| The alpha values and local dimensions have the same range, and f(alpha) is the dimension of the
set with local dimension alpha. That is, f(alpha) = dim(Ealpha). |
| The maximum value of the f(alpha) curve (which occurs at q = 0) is the dimension of the attractor
of the IFS. |
| The f(alpha) curve intersects the line y = x at a single point (corresponding to q = 1) giving a
value called the dimension of the measure. |