Question

Suppose that we pick a point at random from the middle-third Cantor set. What's the probability that this point lies to the left of x, where 0≤x≤1 is some fixed number? Let C be the middle-third Cantor set. Let C₀ be the first set in the construction of C (i.e. C₀ is the interval [0,1]). Let C₁ be C₀ with the middle-third removed. Let C₂ be the set where each segment of C₁ has its middle-third removed. In general, the set Cₙ

is the set given by the nth step of the construction of the Cantor set. See the diagram below. Define a function P(x) in the following way. For a given x,P(x) is the probability that a uniformly randomly chosen point in C is to the left of x. This function is called the devil's staircase. (a) It is easiest to visualize P(x) by building it up in stages. Let P₀ (x) be the probability that a randomly chosen point in C₀ lies to the left of x. Draw the graph of P₀(x) (b) Let P₁(x) be the probability that a randomly chosen point in C₁ lies to the left of x. Draw the graph of P₁(x). (Hint: It should have a plateau in the middle.) (c) Define P₂(x) and P₃(x) analogously but with the sets C₂ and C₃
. Draw the graphs of each and be careful about the widths and heights of the plateaus. (d) The limiting function P(x) is the devil's staircase. Is P(x) continuous? Is P(x) differentiable? Describe the derivative of P. In particular, what is the derivative on any interval that was removed during the construction of the C ?

Answer 1

The function P(x) concerning the Cantor set is a step function that becomes more refined at each stage of the Cantor set construction, resulting in a graph known as the devil's staircase. This function is continuous but not differentiable, and its derivative is mostly zero or undefined.

Step-by-step explanation:

The problem describes a probability function P(x) related to the Cantor set, which is constructed iteratively by removing the middle third of each interval at each stage. Given the way Cantor set is built, the graphs of Pₒ₀(x), Pₒ₁(x), Pₒ₂(x), and Pₒ₃(x) will be step functions that become increasingly complex. Each function prior to the limit describes the uniform distribution probability for the existing segments after n-th iteration.

(a) The graph of Pₒ₀(x) would simply be a straight line from (0,0) to (1,1) since C₀ is the interval [0,1], meaning any point has an equal chance of being to the left of any value x.

(b) For Pₒ₁(x), the graph would show a plateau (horizontal line) between (1/3, 1/2) and (2/3, 1/2) due to the removal of the middle third, with diagonal lines connecting the ends of the plateau to the points (0,0) and (1,1).

(c) Pₒ₂(x) and Pₒ₃(x) are more complex, with additional plateaus corresponding to the segments removed. Each plateau represents a zero-probability region (the segment that was removed), so these regions are 'flat' in the graph.

(d) The limiting function P(x), the devil's staircase, is indeed continuous as it has no gaps or jumps. However, it is not differentiable since the Cantor set constructed is totally disconnected and has no intervals of positive length; it has a countably infinite number of plateaus. The derivative of P(x) exists only on the Cantor set itself where it is zero, and it is undefined on the intervals that were removed, effectively meaning the derivative is most often zero or undefined.