PUMaC 2024 · 加试 · 第 2 题

Problem 2.1.9. Prove that H is simply the counting measure. Because of this, we will freely assume s > 0. 3 Interlude: some fractal constructions There are no problems in this section: instead, we will provide several constructions of fractals which we will base problems off of in later sections. 3.1 Cantor Set 1 2 Let C be the closed unit interval C = [0 , 1] ⊂ R . Then let C = C \ ( , ) = 0 0 1 0 3 3 1 2 [0 , ] ∪ [ , 1]. Notice that C is the disjoint union of two closed intervals. We will construct 1 3 3 C from C as follows: by induction, each C is a finite disjoint union of closed intervals n +1 n n S − n C = [ a , b ], and each interval is length 3 . Thus, let n j j j [ − n − 1 − n − 1 C = [ a , b ] \ ( a + 3 , b − 3 ) n +1 j j j j j This process removes the middle third of each interval. Since C is clearly also the n +1 finite union of disjoint closed intervals, it follows by induction that C exists for every i , i and that C ⊂ C . i +1 i Then, define the Cantor Set C as ∞ C = ∩ C i i =0 This intersection is non-empty and compact since it is the intersection of a decreasing sequence of compact sets. 3.2 Sierpinski Carpet Figure 1: The construction of the Sierpinski Carpet. Start with the closed unit square □ = [0 , 1] × [0 , 1]. To go from □ to □ , divide 0 i i +1 each component square into 9 equal new squares, and remove the central open square, as k − k in figure 1. At stage □ there will be 8 squares, each with sidelength 3 . k Note that when we remove the center square, we are deleting the open component. 1 2 1 2 For example, □ = □ \ ( , ) × ( , ) . Thus, each □ is closed and so compact, and 1 0 i 3 3 3 3 □ ⊂ □ . Thus, i +1 i
□ = □ i i Is non-empty by compactness. □ is the Sierpinski Carpet. 3.3 Minkowski Sausage Let M = [0 , 1] the interval. At each stage, replace every interval in M with the interval 0 i − k pattern as in figure 2 on the left to create M . Each interval in M is length 4 , and there i +1 k k are 8 such intervals. Notice that in M , for example, we consider there to be 8 intervals 1 1 each with length , even though two of the intervals are colinear. There is a well-defined 4 limit of this process which results in a compact fractal denoted as M , called the Minkowski Sausage. 3.4 Koch Curve Let K = [0 , 1] the interval. At each stage, replace every interval in K with the interval 0 i − k pattern as in figure 2 on the right to create K . Each interval in K is length 3 , and i +1 k k there are 4 such intervals. Notice that in K , for example, we consider there to be 4 1 1 intervals each with length . There is a well-defined limit of this process which results in a 3 compact fractal denoted as K , called the Koch Curve. Figure 2: (Left) the construction of the Minkowski Sausage. (Right) the Koch Curve 4 Iterated Function Systems An iterated function system is a method of generating self-similar fractals. All the fractals in the interlude, as well as many more, may be created using this method. n n Definition 4.0.1. A function f : R → R is called a contraction if there exists some n constant 0 < c < 1 such that for every x, y ∈ R , | f ( x ) − f ( y ) | = c | x − y | . The value c is called the contraction ratio for f . Reworded, a contraction is a function that sends points closer to each-other by some common factor. An iterated function system ( IFS ), then, is just a finite set of contractions ( f , ..., f ), 1 m m ≥ 2 (they are assumed to all have the same domain and codomain, but not necessarily the same contraction ratios). n ( k )