Multifractals

The range of alpha values

Click each equal sign for an explanation of that equality or inequality.

Say ln(p_a)/ln(r_a) = min{ln(p_i)/ln(r_i)}. Then for all i, ln(p_i)*ln(r_a) >= ln(p_a)*ln(r_i)

Now alpha = -dtau/dq =
( p_i^qr_i^tau(q)(ln(p_i))/( p_i^qr_i^tau(q)(ln(r_i)) =

(( p_i^qr_i^tau(q)(ln(p_i))/( p_i^qr_i^tau(q)(ln(r_i))) * (ln(r_a)/ln(p_a)) * (ln(p_a)/ln(r_a)) =

(( p_i^qr_i^tau(q)(ln(p_i)*ln(r_a))/( p_i^qr_i^tau(q)(ln(r_i)*ln(p_a))) * (ln(p_a)/ln(r_a)) >=

(( p_i^qr_i^tau(q)(ln(p_a)*ln(r_i))/( p_i^qr_i^tau(q)(ln(r_i)*ln(p_a))) * (ln(p_a)/ln(r_a)) =

ln(p_a)/ln(r_a)

The inequality alpha <= max{ln(p_i)/ln(r_i)} is handled similarly.

Combining these bounds we obtain

alpha_min <= alpha <= alpha_max

Return to Multifractals from IFS.