CHAPTER 2 Dimension Since the notion of a tangent plane is absent in the fractal setting, we need an alternative way to explain why a curve has dimension 1 and a surface has dimension 2. We will now discuss a construction that works in any compact metric space X. Let A be a closed subset of X. Fix some radius r 0. Let Nr(A) be the least number of balls of radius r needed to cover A. If A is a rectifiable curve in Rn, it is not hard to show that Nr(A) is of order r−1. If A is a surface, Nr(A) is approximately r−2. This suggests the idea to define the dimension of an arbitrary set as the number d for which Nr(A) r−d. Definition. The limit lim r→0+ log Nr(A) log 1/r , when it exists, is called the Minkowski dimension of A and is denoted by M-dimA. Even if the limit fails to exist, we still can talk about the upper and the lower limits, which give us the upper and the lower Minkowski dimensions M-dimA = lim sup r→0+ log Nr(A) log 1 r and M-dimA = lim inf r→0+ log Nr(A) log 1 r correspondingly. Since Nr(A) is a non-increasing function of r and since log 1 r changes very slowly, instead of the (upper or lower) limit along all values of r we can take the (upper or lower) limit along any sequence rj such that log 1 rj+1 log 1 rj 1 as j ∞. When X = Q is the unit cube in Rn, we can use cubic boxes instead of balls. More precisely, take some integer p 0 and split Q into pn equal subcubes with sidelength 1 p . For A Q, let Hp(A) be the number of subcubes that intersect A. Definition. The box dimension of A is B-dimA = lim p→+∞ log Hp(A) log p . Again, in the cases when the limit fails to exist, we may talk about the upper and the lower box dimensions instead. Since each cube with sidelength 1 p is contained in a ball of radius n p , say, we conclude that N n p (A) Hp(A). Since each ball of radius 1 p can intersect at most some finite number C(n) cubes of size 1 p , we have Hp(A) C(n) N 1 p (A). Taking the logarithm, dividing by log p, and passing to the limit as p +∞, we see that the box dimension we just defined coincides with the Minkowski dimension and the same is true for the upper and lower versions. 11
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