Multifractals

Limiting values of tau(q)

The function tau(q) is determined by the equation

(p1q)(r1tau(q)) + ... + (pNq)(rNtau(q)) = 1 (*)

Recall that 0 < ri < 1 and 0 < pi < 1 for all i.

As q -> infinity, each piq -> 0, and so at least one of the ritau(q) must become arbitrarily large if equation (*) is to hold.

In order for any of the ritau(q) to become arbitrarily large, we must have tau(q) -> -infinity.

As q -> -infinity, each piq -> infinity, and so each ritau(q) must go to 0 if equation (*) is to hold.

In order for ritau(q) -> 0, we must have tau(q) -> -infinity.

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