We consider the functions |
p1qr1tau(q) + ... + pNqrNtau(q) = 1 |
can be written |
h1(q) + ... + hN(q) = 1. |
Differentiating, |
dhi/dq = piqritau(q) (ln(pi) + ln(ri)(dtau/dq)) |
Suppose |
Recalling alpha = -dtau/dq, the bounds
|
-ln(pm)/ln(rm) >= dtau/dq >= -ln(pM)/ln(rM) |
Because |
-ln(ri)(ln(pm)/ln(rm)) <= ln(ri)(dtau/dq) <= -ln(ri)(ln(pM)/ln(rM)) |
Adding ln(pi) to each side |
ln(pi) - ln(ri)(ln(pm)/ln(rm)) <= ln(pi) + ln(ri)(dtau/dq) <= ln(pi) - ln(ri)(ln(pM)/ln(rM)) |
Taking i = m
gives |
For all i, |
Because hm(q) is a nondecreasing function bounded between 0 and 1, both limits |
limq -> infinityhm(q) and limq -> -infinityhm(q) |
exist. These limits exist also for |
Moreover, |
In fact, each dhi/dq has
at most one zero,
and so the limits |
If |
Here is a picture of some typical hi. |
From this we
deduce
|
Because dtau/dq appraoches |
Return to Multifractals from IFS.