Australian National University Student Number: u Mathematical Sciences Institute EXAMINATION: Semester 1 — Final Exam, 2020 MATH3062/6116 — Fractal Geometry and Chaotic Dynamics Exam Duration: Exam due via Wattle on Wednesday 24 June at 11:55pm Reading Time: N/A Materials Permitted For The Exam: • Unmarked English-to-foreign-language dictionary (no approval from MSI required), ocial course textbooks, any resources posted to Wattle, your own solutions to the homeworks/quizzes/labs/midsemester exam, any personal notes you have taken for the course. Materials To Be Supplied To Students: • N/A Instructions To Students: • You are not allowed to collaborate with your classmates. No online resources are permitted besides those on Wattle. • Non-HPO students solve problems 1 through 6. HPO students solve all problems except problems 1 and 6. • You total exam mark will be out of 200 points. • Provide full justication for your answers when requested. Q1 30 Q2 35 Q3 35 Q4 35 Q5 35 Q6 30 Q7 30 Q8 30 Total Question 1 30 pts (a) Let C denote the standard Cantor set. That is, C is the attractor of the IFS,{ [0, 1];w1(x) = 13x, w2(x) = 1 3x + 2 3 } . For arbitrary S ⊆ R , put 3S = {3x : x ∈ S} . Prove that 3 ( C ∩ [ 0, 13 ] ) = C . 8 pts (b) Let (X ,d) be a complete metric space. Suppose f : X → X is a contraction mapping with contractivity factor 0 ≤ s < 1 . Denote by x f the unique xed point of f . Prove that for all x ∈ X , d(x, x f ) ≤ (1 − s)−1d(x, f (x)). See Lemma 11.1 in Chapter 3 of FE. You must write the solution in your own words. 10 pts (c) Let (X ,d) be a metric space. For any x ∈ X and ε > 0 , let Bε(x) = {y ∈ X : d(x,y) < ε} . Let D be a subset of X . Dene what is means for D to be dense in X . You may give any one of several equivalent denitions. 4 pts (d) Let (Σ,dΣ) be the code space on the symbols {1, 2, . . . ,N } , where dΣ is the usual metric dΣ(σ ,ω) = ∞∑ i=1 |σi − ωi | (N + 1)i . Let f denote the "left-shift map" on Σ : f (σ ) = f (σ1,σ2,σ3, . . . ) = (σ2,σ3, . . . ). Recall that in Workshop 5 we constructed an element ω ∈ Σ that has a dense orbit. A transformation д : Σ → Σ is called transitive if, given any σ , σ˜ ∈ Σ and ε, ε˜ > 0 , there exists N ∈ N such that д◦N (A) ∩B , ∅ , where A = Bε(σ ) , B = Bε˜(σ˜ ) . Using the element ω mentioned above, explain why the left-shift map f : Σ→ Σ is transitive. You do not need to give a formal proof, but you can if you want to. Hint: Given any σ ∈ Σ , how can you change nitely many terms of ω to make ω close to σ ? 8 pts MATH3062/6116 — S1, Page 2 of 25 Write your solution here MATH3062/6116 — S1, Page 3 of 25 Extra space for previous question MATH3062/6116 — S1, Page 4 of 25 Question 2 35 pts (a) Let (X ,d) be a metric space and f : X → X a transformation. Suppose x f is a xed point of f . Dene what is means for x f to be attractive. Dene what it means for x f to be repelling or repulsive. 4 pts (b) Consider the family of polynomials Qc(x) = x2 + c , x, c ∈ R . Prove the following. (i) If c > 1/4 , Qc has no xed points. 4 pts (ii) If c = 1/4 , Qc has one xed point x1 . Find its value. Show that if x0 < x1 and x0 is suciently close to x1 , then Q◦kc (x0) → x1 . On the other hand, if x0 > x1 , show that Q◦kc (x0) → ∞ (Hint : use the monotone convergence theorem). 6 pts (iii) if c ∈ (−3/4, 1/4) , Qc has one attractive xed point x1 and one repulsive xed point x2 . Find their values. 5 pts (iv) If c < −3/4 , Qc has two repulsive xed points x1 and x2 . Find their values. 5 pts For (iii) and (iv), you may use without proof the fact we discussed in lecture, that the value of | f ′| can, in some cases, allow you to conclude that a xed point is attractive or repulsive. (c) Consider the logistic map fλ : R→ R , fλ(x) = λx(1 − x) , λ > 1 . Prove the following. (i) If 1 < λ ≤ 3 , then fλ has no points of minimal period two. 5 pts (ii) If λ > 3 , then fλ has exactly two points x1 , x2 , of minimal period two. Find their values. 6 pts Write your solution here MATH3062/6116 — S1, Page 5 of 25 Extra space for previous question MATH3062/6116 — S1, Page 6 of 25 Extra space for previous question MATH3062/6116 — S1, Page 7 of 25 Question 3 35 pts (a) Dene what it means for an m × m matrix O : Rm → Rm to be an orthogonal transformation. Dene what it means for S : Rm → Rm to be a similitude with scaling factor 0 < r < 1 . Note that m ∈ N is arbitrary. 5 pts (b) Let X ⊆ Rm be nonempty, closed, and equipped with the Euclidean metric. Let {X ;w1,w2, . . . ,wN } be an IFS, where each wi : X → X is a similitude with scaling factor 0 < r < 1 . Dene the Hutchinson operator W associated to this IFS. 4 pts (c) Dene what it means for an open subset U of Rm with U ⊆ X to satisfy the just- touching or separating set condition with respect to the above IFS. 4 pts (d) Let A be the attractor of the above IFS. If this IFS admits a separating open subset U , give a formula for the fractal dimension D(A) of A . 4 pts (e) Let X be the closed unit square [0, 1] × [0, 1] in R2 . consider the IFS {X : fj, j = 1, . . . , 8} , where A = [ 1/3 0 0 1/3 ] and f1(x) = Ax, f2(x) = Ax + [ 0 1/3 ] , f3(x) = Ax + [ 0 2/3 ] , f4(x) = Ax + [ 1/3 0 ] , f5(x) = Ax + [ 1/3 2/3 ] , f6(x) = Ax + [ 2/3 0 ] , f7(x) = Ax + [ 2/3 1/3 ] , f8(x) = Ax + [ 2/3 2/3 ] . Draw two pictures, the rst one depicting a separating open set U , and the second depicting the eight images of U under the maps fj . Label the images Uj , corresponding to the fj they come from. You do not need to give a formal proof. Use your formula from (d) to give an exact answer for the fractal dimension of the attractor of this IFS. 8 pts (f) The attractor of the IFS from (e) is known as the Sierpinski carpet, which we denote by C . Modify your Lab 3 code so it takes as the partial address [2, 5, 4] and highlights in red all points in C that have those digits to start their address. Submit your code with the exam, as .pdf or .txt le—you do not need to make comments. Your code should produce exactly the four gures below. Submit copies (.jpeg or .png) of these gures which your code produced. Use the following RGB array to assign colors to the maps fj . The rst color corresponds to f1 , the second to f2 , and so forth: 10 pts colours = [[255, 255, 0], [0, 255, 0], [0, 0, 255], [0, 127, 0], [255, 127, 0], [255, 127, 127], [0, 0, 127], [0, 127, 127]]. MATH3062/6116 — S1, Page 8 of 25 MATH3062/6116 — S1, Page 9 of 25 Write your solution here MATH3062/6116 — S1, Page 10 of 25 Extra space for previous question MATH3062/6116 — S1, Page 11 of 25 Question 4 35 pts Fix r ∈ R . Suppose f : R→ R is a degree three polynomial such that f (r ) = f ′(r ) = 0 , and f ′′(r ) , 0 . That is f (x) = a(x − r )2 + b(x − r )3, for some a,b ∈ R with a, b , 0 . (a) Prove there exists δ0 > 0 suciently small so that 0 < |x0−r | < δ0 implies f ′(x0) , 0 . You do not need to explicitly nd δ0 . 10 pts (b) Let δ0 > 0 be as you found in part (a). Consider the Newton iteration x0 = initial guess, xn+1 = xn − f (xn) f ′(xn), en = xn − r . Show there exists 0 < δ ≤ δ0 so that 0 < |x0 − r | ≤ δ implies f ′(xn) , 0, n = 0, 1, 2, . . . , (∗) |en+1 | ≤ 34 |en |. (∗∗) You do not need to explicitly nd δ . 10 pts (c) Let δ be as you found in part (b). Use (∗) and (∗∗) to prove that, if 0 < |x0 − r | < δ , then |en | → 0 as n →∞ . 5 pts (d) Using the conclusion of Example 1 from Lecture 21, nd a value of c so that the Julia set of the Newton iteration z 7→ z − д(z) д′(z), д(z) = z 2 − c, is the line y = (−4/3)x . Justify your answer. 10 pts Write your solution here MATH3062/6116 — S1, Page 12 of 25 Extra space for previous question MATH3062/6116 — S1, Page 13 of 25 Extra space for previous question MATH3062/6116 — S1, Page 14 of 25 Question 5 35 pts Let (X ,d) be a metric space. Let f (z) = anzn + an−1zn−1 + · · · + a0 be a complex polynomial of degree n ≥ 2 . (a) Let S ⊆ X be nonempty. Dene the boundary ∂S of S . 5 pts (b) Dene the lled-in Julia set K(f ) and the Julia set J (f ) of f . 5 pts (c) What are the invariance properties of the J (f )? 5 pts (d) Let α, β, c ∈ C , α , 0 , and put h(z) = αz + β , fc(z) = z2 + c . Dene f = h−1 ◦ fc ◦ h. Using the fact we proved on page 12 of Lecture 21, a give formal set containment argument to prove K(f ) = h−1(K(fc)). (!) 10 pts (e) Using (!), nd a, b, c′ ∈ C so that the polynomial f (z) = az2 + bz + c′ has lled in Julia set the closed disc of radius 1/2 centered at 1 + i (clearly indicate what a, b, c′ you nd). Modify your Lab 5 code to produce exactly the following output below. Submit your code (.pdf or .txt, you do not need to make comments) and your image (.jpeg or .png) with the exam. 10 pts MATH3062/6116 — S1, Page 15 of 25 Write your solution here MATH3062/6116 — S1, Page 16 of 25 Extra space for previous question MATH3062/6116 — S1, Page 17 of 25 Question 6 30 pts This question is a version of the Workshop 10 problem. Let f be a complex polynomial of degree n ≥ 2 . Fact. If z ∈ J (f ) , then J (f ) is the closure of ∪∞ k=1(f ◦k)−1({z}) . Theorem. If f is a polynomial, J (f ) is the closure of the repelling periodic points of f . (a) Let (X ,d) be a metric space. Dene what it means for a subset A of X to be connected. 5 pts (b) Dene the Mandelbrot set of the family of quadratic polynomials fc(z) = z2 + c, c ∈ C. 5 pts (c) State the characterization of the Mandelbrot set in terms of orbits of z = 0 under fc . Use it prove that the Julia set of the polynomial f (z) = z2 − 2 is connected. 8 pts (d) Use the above Fact and Theorem to show {2,−2} ⊆ J (z2 − 2) ⊆ [−2, 2]. 12 pts Write your solution here MATH3062/6116 — S1, Page 18 of 25 Extra space for previous question MATH3062/6116 — S1, Page 19 of 25 Question 7 30 pts For this question, it will be helpful to refer to the excerpt from Folland’s analysis textbook that is available on Wattle. (a) For A ⊆ Rn , s ≥ 0 , and δ > 0 given, we dened, in HPO Lecture 5, the quantity H s δ (A) , using countable covers of A by arbitrary subsets of diameter ≤ δ . Dene an analogous quantity, but this time for A ⊆ X , where (X ,d) is a general metric space. 5 pts (b) Using your answer to part (a), dene, as a certain limit as δ → 0 , the s -dimensional Hausdor measure H s(A) , for A a subset of a general metric space (X ,d) . Justify why this limit always exists, as either a nite nonnegative number or ∞ . 5 pts (c) The following proposition is proved in the Folland excerpt I provided, but you must write the proof in your own words. Let s ≥ 0 , let (X ,d) be a metric space, and let f , д be functions from some set Y into X . Suppose further that there exists C > 0 so that for all y, z ∈ Y , d(f (y), f (z)) ≤ Cd(д(y),д(z)). Prove that H s(f (A)) ≤ CsH s(д(A)) . 10 pts (d) Let A be a Borel subset of Rn (this is only a technical assumption and not key to the proof). Let Ln denote Lebesgue measure as dened in HPO Lecture 5. Suppose Hn(A) = 0 . Prove that Ln(A) = 0 . You will need to use (without proof) that any (open or closed) ball Br ⊆ Rn of radius r > 0 has Ln(Br ) = cnrn for a constant cn > 0 depending only on n . 10 pts Write your solution here MATH3062/6116 — S1, Page 20 of 25 Extra space for previous question MATH3062/6116 — S1, Page 21 of 25 Extra space for previous question MATH3062/6116 — S1, Page 22 of 25 Question 8 30 pts (a) Let U ⊆ C be open and suppose д : U → C . Dene what is means for д to be analytic or holomorphic on U . 5 pts (b) Let {дk}∞k=1 be a sequence of analytic functions on U . Dene what it means for the family to be normal at a point w ∈ U . 5 pts (c) State Montel’s Theorem. 5 pts (d) Let f be a complex polynomial of degree n ≥ 2 . Let ω (allowing ω = ∞) be an attractive xed point of f . Rewrite, in your own words, the proof of Lemma FG 14.11 from the HPO Lecture 11 slides. That is, prove that the boundary of the basis of attraction of ω is the Julia set of f . It is very important that you rewrite the proof in your own words, otherwise you will get no points. Take care to ll in the details I left out. You may quote without proof the same theorems I do. In particular, you may use: If U is a connected open subset of the complex plane and д : U → C is analytic on U , then if д is identically a constant on some open subset of U , then д is identically constant on all of U . 15 pts Write your solution here MATH3062/6116 — S1, Page 23 of 25 Extra space for previous question MATH3062/6116 — S1, Page 24 of 25 Extra space for previous question MATH3062/6116 — S1, Page 25 of 25





















































































































































































































































































































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