4 HILLEL FURSTENBERG iterations of simple classical processes. Consider, for instance, the standard middle- third Cantor set on the line. Its construction starts with a closed interval, say [0, 1]: C0 Figure 5. The construction of the Cantor set starts with a single interval. At the first step, the middle third ( 1 3 , 2 3 ) is removed and we get the set C1 = [0, 1 3 ] ∪ [ 2 3 , 1]: C1 Figure 6. At the first step the middle third is removed. Next, the same procedure of removing the middle third is applied to each of the intervals of C1 resulting in 4 intervals of length 1 9 : C2 Figure 7. This procedure is repeated for each remaining interval. This process repeats again and again, so Cn consists of 2n intervals of length 3−n. The Cantor set C is defined as the limit of Cn in the Hausdorff metric, which in this case is the same as ∞ n=1 Cn. Note that the Lebesgue measure of Cn equals ( 2 3 )n, which tends to 0 as n → ∞, so we remove almost all points from the initial interval [0, 1] in the sense of the Lebesgue measure. However, there are still many points left. For instance, all endpoints of the intervals appearing at all stages stay, so C is infinite. It is possible to show that C is of the cardinality of the continuum, so from this point of view, it is as large as the original interval [0, 1]. The Sierpinski gasket S is just a 2-dimensional version of the middle third Cantor set. We start with with a closed equilateral triangle, split it into 4 equal triangles and remove the central one: Figure 8. At the first step, the central triangle is removed. Then we repeat this procedure for each of the three remaining triangles: and so on. After n steps, we get 3n triangles with side-length 2−n. Again, the area of Sn decays as a geometric progression. However, the total length ( 3 2 )n of the sides

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2014 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.