Multifractals

Unique value for tau(q)

To see that for each q the equation (p₁^q)(r₁^tau(q)) + ... + (p_N^q)(r_N^tau(q)) = 1 determines a unique value of tau(q), for each q define a function g(tau) by

g(tau) = (p₁^q)(r₁^tau) + ... + (p_N^q)(r_N^tau)

Certainly, g(tau) is a continuous function.

Recall that 0 < r_i < 1 and 0 < p_i < 1 for all i.

As tau -> infinity, each r_i^tau -> 0, and so g(tau) -> 0.

As tau -> -infinity, each r_i^tau -> infinity, and so g(tau) -> infinity.

Finally, dg/dtau = (p₁^q)(r₁^tauln(r₁)) + ... + (p_N^q)(r_N^tauln(r_N)) < 0

because each ln(r_i) < 0.

Combining these three observations, we see the graph of g(tau) must look something like this

That is, g(tau) decreases from large positive values to nearly 0, so g(tau) = 1 for exactly one value of tau, and this is tau(q) for the given value of q.

If all the r_i take on a common value, r, the equation

(p₁^q)(r₁^tau(q)) + ... + (p_N^q)(r_N^tau(q)) = 1

can be solved for tau(q):

tau(q) = -ln(p₁^q + ... + p_N^q)/ln(r).