Multifractals from IFS

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.

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