Multifractal analysis

Yesterday, the Edinburgh Mathematical Society met in St Andrews. Thomas Jordan gave a talk on multifractal analysis. I had no idea what this was, and went along without too much expectation of being enlightened. But Thomas began with a well-chosen example (dating from before the term “multifractal analysis” was invented) that gave the flavour very clearly. So here is a summary of the first ten minutes or so of the talk.

Any real number x in [0,1] has a base 2 expansion. The density of 1s in this expansion (the limit of the ratio of the number of 1s in the first n digits to n) need not exist; but, according to the Strong Law of Large Numbers, for almost all x, the limit exists, and is 1/2.

So it might seem perverse to consider the set Xp of numbers for which the limit is equal to p, for arbitrary p. But that is precisely what Besicovich did, and he proved a nice theorem about the Hausdorff dimension of this set.

Briefly, given a subset of [0,1], take a cover of it by intervals of length at most δ, and take the sum of the sth powers of the lengths of the intervals. Now take the infimum of this quantity over all such coverings, and the limit of the result as δ tends to 0. The result is the s-dimensional Hausdorff measure of the set. There is a number s0 such that the measure is ∞ for s < s0, and is 0 for s > s0; for s = s0, it may take any value. This s0 is the Hausdorff dimension of the set.

Besicovich calculated the Hausdorff dimension of the sets Xp defined earlier. It turns out to be equal to the binary entropy function of p; this is H(p) = −p log p−(1−p)log (1−p), where the logarithms are to base 2. The function H(p) takes its maximum value when p = 1/2, when it is equal to 1; and 1-dimensional Hausdorff measure coincides with Lebesgue measure. So Besicovich’s result handles this case correctly.

The entropy function suggests a relation to binomial coefficients and Stirling’s formula, which is indeed involved in the proof. (The logarithm of the binomial coefficient {n choose pn} is asympototically nH(p), as follows easily from Stirling’s approximation for n!.)

All this can be phrased in terms of the dynamics of the map 2x mod 1 on the unit interval (which acts as the left shift on the base 2 expansion), which suggests a good direction for generalisation, and suggests too that that this generalisation will involve concepts from ergodic theory such as entropy and pressure. Most of the lecture was about this.

(Perhaps worth mentioning that all these sets are negligible in the sense of Baire category, according to which almost all real numbers have the property that the lim inf of the density is 0 and the lim sup is 1.)

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About Peter Cameron

I count all the things that need to be counted.
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