| On the left we have a graph of the probabilities, on the right the
coarse Holder exponents. How can we explain the different directions of
curvature? |
|
| For example, the corner with address 111... has the lowest measure
but the highest coarse Holder exponent, while the corner with address 444... has the highest
measure but the lowest coarse Holder exponent. |
| This is an algabraic consequence of the fact that numbers
between 0 and 1 have negative logarithms. |
| The square with address the string 111...1 of
n 1s has probability p1n and side length 0.5n. |
| Similarly, the square
with address the string 444...4 of
n 4s has probability p4n and side length 0.5n. |
| The first has coarse Holder exponent |
| limn -> infinity Log(p1n) / Log(0.5n) |
| = limn -> infinity (n*Log(p1)) / (n*Log(0.5)) |
| = limn -> infinity Log(p1) / Log(0.5) |
| = Log(p1) / Log(0.5) |
|
| The second has coarse Holder exponent Log(p4) / Log(0.5). |
| To compare these, recall Log is an increasing function, so |
| p1 < p4 implies Log(p1) < Log(p4) |
| Next, the scaling ratios r lie between 0 and 1, so Log(r) < 0. In this example,
Log(0.5) < 0. Consequently, |
| Log(p1) / Log(0.5) > Log(p4) / Log(0.5). |
