M3A23/M4A23. Specimen Paper

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1 UNIVERSITY OF LONDON Course: M3A23/M4A23 Setter: J. Lamb Checker: S. Luzzatto Editor: Editor External: External Date: March 26, 2009 BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2008 M3A23/M4A23 Specimen Paper Setter s signature Checker s signature Editor s signature

2 UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2008 This paper is also taken for the relevant examination for the Associateship. M3A23/M4A23 Specimen Paper Date: examdate Time: examtime Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used.

3 1. Let X be a metric space with metric (distance function) d : X X R. List the properties that d must satisfy in order to be a metric. (b) Give the definition of a contraction F : X X, with respect to the metric d. (d) (e) For each of the combination of properties listed below of a contraction F on a metric space X, verify whether the situation can arise, and if so give one explicit example of F : X X that realizes the listed properties: (i) F has a unique fixed point and X is complete. (ii) F has no fixed point and X is complete. (iii) F has a unique fixed point and X is not complete. (iv) F has no fixed point and X is not complete. Prove the following variant of the Contraction Mapping Theorem: Theorem. Let F : X X be a Lipschitz map on a complete metric space X. Suppose that F is not a contraction, but F m is a contraction for some m > 1. Then F has a unique fixed point and all x X converge to this fixed point with exponential speed. Give an example of a map F that satisfies the conditions set out in the Theorem stated under (d). 2. Give a definition of topological mixing for a circle map F : S 1 S 1. (b) (d) (e) Show that invertible circle maps are never topologically mixing. [You may use results that were derived in the course without proof, but any such results should be stated carefully.] Show that all expanding circle maps are topologically mixing. Describe the condition on the parameters a, b and c that must be satisfied for F (x) = (ax + b cos 2 (2πx) + c) mod 1 to be an orientation preserving and expanding circle map. Show that if the circle map F of part (d) is orientation preserving and expanding then F k has a k 1 fixed points. Show that this implies that F has periodic orbits with period k for all k Z +.

4 3. Let F : S 1 S 1 defined by F (x) = (ax + b cos 2 (2πx) + c) mod 1 with x [0, 1) = S 1 diffeomorphism. and parameters a, b, c R be an orientation preserving circle (b) (d) (e) Determine what conditions on the parameters a, b, and c must be satisfied for F to be an orientation preserving circle diffeomorphism. Give a formula for the rotation number ρ(f ) of F. Show that although this formula refers to a specific point x S 1, ρ(f ) is independent of the choice of this point x. Prove the following proposition: Proposition. If ρ(f ) is irrational then for all x S 1 and any m, n N with m > n the following holds: for every y S 1 the forward orbit {F k (y) k N} intersects the interval I = [F m (x), F n (x)]. [You may use without proof results about properties of circle maps with rational rotation number, but any such result should be stated carefully.] Give the definition of the ω-limit set ω(x) for the map F and show that ω(x) is independent of x if ρ(f ) is irrational. [Hint: use the Proposition stated in part of this question.] Discuss the structure of the ω-limit set of F when ρ(f ) is irrational. [You may use results from the course without proof, but any such results should be stated carefully.]

5 4. Consider the cubic map with λ > 0. f λ : R R, x λx(1 x 2 ), For small λ > 0, x = 0 is an asymptotically stable fixed point. Determine the parameter value at which this fixed point loses stability. Determine the type of local bifurcation that occurs. Is it a generic (typical) local bifurcation of one-dimensional maps with one parameter? [Motivate your answer.] (b) (i) Find the smallest value of λ > 0 at which a periodic doubling bifurcation occurs (from a fixed point). Note that at this parameter value two period doubling bifurcations occur at the same time. Verify that the branches of period two orbits have the explicit expression ±1 λ = x 1 x. 2 (ii) (iii) Show that in a typical period doubling bifurcation, the amplitude of the period doubled orbit grows as a the square root of the bifurcation parameter. Verify whether such growth is also observed in the period doubling bifurcation of f λ in part (i) of this question. Discuss how the map f λ in the neighbourhood of a period-doubling bifurcation point can be considered as the Poincaré return map of a flow on a two-dimensional surface. Describe (and sketch!) the local geometry of this suspension flow and illustrate the period-doubling bifurcation on it (in pictures). Determine for which values of the parameter λ, the map f λ restricted to the set of initial conditions for which orbits do not escape from [0, 1] is topologically conjugate to a full shift on two symbols. [You do not need to prove the conjugacy, but need to show by what criteria you determine λ and how the conjugacy is defined.]

6 5. Let Σ 3 denote the set of infinite sequences {ω i } i N whose entries ω i are taken from a set of three symbols {1, 2, 3}. (Σ 3, d µ ) is a metric space with d µ (ω, ω ) := m N δ(ω m, ω m) µ m, where µ > 1 and δ(a, b) = 0 if a = b and δ(a, b) = 1 if a b. Determine for which values of µ R the cylinder C α0,...,α n 1 := {ω Σ 3 ω i = α i, 0 i < n}. is the open ball in Σ 3 around α = α 0... α n 1... with radius µ 1 n. (b) Consider the piecewise linear map F : [0, 1] [0, 1] given by 2x if x I 1 F (x) = 3 4x if x I 2 2x 3 if x I 2 3 where I 1 = [0, 1 2 ], I 2 = [ 1 2, 3 4 ], and I 3 = [ 3 4, 1]. (i) Show that the dynamics of this map is semi-conjugate to a three-state Markov chain with a Markov graph as given in the figure that is depicted below, where points x [0, 1] are coded as ω Σ 3 with ω n = k if F n (x) I k. [Hint: Show that I ω0...ω n 1 ( 2) n, where I ω0...ω n 1 denotes the set of initial conditions in [0, 1] that are represented by a symbolic string in Σ 3 starting with ω 0... ω n 1.] (ii) Determine whether the following statements are correct the Markov chain in (i) is topologically transitive F is topologically transitive Motivate your answer. [You may use results from the course without proof, but any such results should be stated carefully.] (iii) Show that F 4 has 34 fixed points by a different method than writing out all admissible periodic words of length 4 for the Markov chain.

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