Multifractals

Limiting values of β(q)

The function β(q) is determined by the equation

(p1q)(r1β(q)) + ... + (pNq)(rNβ(q)) = 1 (*)

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

As q → ∞, each piq → 0, and so at least one of the riβ(q) must become arbitrarily large if equation (*) is to hold.

In order for any of the riβ(q) to become arbitrarily large, we must have β(q) → -∞.

As q → -∞, each piq → ∞, and so each riβ(q) must go to 0 if equation (*) is to hold.

In order for riβ(q) → 0, we must have β(q) → ∞.

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