Multifractals

Asymptotes of the tau(q) curve

We consider the functions h_i(q) = p_i^qr_i^tau(q) for i = 1, ..., N. These are natural functions because the equation defining tau(q)

p₁^qr₁^tau(q) + ... + p_N^qr_N^tau(q) = 1

can be written

h₁(q) + ... + h_N(q) = 1.

Differentiating,

dh_i/dq = p_i^qr_i^tau(q) (ln(p_i) + ln(r_i)(dtau/dq))

Suppose ln(p_m)/ln(r_m) = alpha_min and ln(p_M)/ln(r_M) = alpha_max

Recalling alpha = -dtau/dq, the bounds alpha_min <= alpha <= alpha_max become

-ln(p_m)/ln(r_m) >= dtau/dq >= -ln(p_M)/ln(r_M)

Because ln(r_i) < 0 for each i, we have

-ln(r_i)(ln(p_m)/ln(r_m)) <= ln(r_i)(dtau/dq) <= -ln(r_i)(ln(p_M)/ln(r_M))

Adding ln(p_i) to each side

ln(p_i) - ln(r_i)(ln(p_m)/ln(r_m)) <= ln(p_i) + ln(r_i)(dtau/dq) <= ln(p_i) - ln(r_i)(ln(p_M)/ln(r_M))

Taking i = m gives dh_m/dq >= 0; taking i = M gives dh_M/dq <= 0.

For all i, 0 <= h_i(q) <= 1 for all q.

Because h_m(q) is a nondecreasing function bounded between 0 and 1, both limits

lim_{q -> infinity}h_m(q) and lim_{q -> -infinity}h_m(q)

exist. These limits exist also for h_M(q).

Moreover, lim_{q -> -infinity}h_M(q) = c_M and lim_{q -> infinity}h_m(q) = c_m, with both c_M and c_m positive.

In fact, each dh_i/dq has at most one zero, and so the limits lim_{q -> infinity}h_i(q) and lim_{q -> -infinity}h_i(q) exist

If lim_{q -> infinity}h_i(q) = c_i > 0, then ln(p_i)/ln(r_i) = ln(p_m)/ln(r_m).

Here is a picture of some typical h_i.

From this we deduce lim_{q -> infinity}dtau/dq = -alpha_min and similarly lim_{q -> -infinity}dtau/dq = -alpha_max

Because dtau/dq appraoches -alpha_max as q -> -infinity and -alpha_minas q -> infinity, we deduce the graph of tau(q) has oblique asymptotes of these slopes.

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