ADDITIONAL PROBLEMS FOR

DYNAMICAL SYSTEMS: STABILITY, SYMBOLIC DYNAMICS, AND CHAOS

SECOND EDITION

by Clark Robinson

p. 63 Extra Problem 2.32. Assume F is a lift of f. Prove that F^{q} is a lift of f^{q}.

p. 63 Extra Problem 2.33. Assume that F_{1} and F_{1} are lifts of f. Prove that there is an integer p such that F_{2}(x) = F_{1}(x) + p for all x.

p. 63 Extra Problem 2.34. Assume that f on X and g on Y are semi-conjuage via the map h,

h ∘ f = g ∘ h.

(a) If p is a periodic point for f with period n, prove that h(p) is a periodic point for g whose period divides n. If f and g are conjuage, prove the periods of p and h(p) are the same.

(b) Assume p is an asymptotically stable fixed point for f. Prove that h(p) is an asymptotically stable fixed point for f.

(c) Assume that f and g are conjugate and p is a fixed point for f with f′(p) > 0. Prove that h(p) is a fixed point with g′(h(p)) ≥ 0.

(d) Prove that if f is topologically transitive on X, then g is topologically transitive on Y.

(e) Prove that if f is topologically mixing on X, then g is topologically mixing on Y.

p. 63 Extra Problem 2.35. Let f : S^{1} → S^{1} be an orientation preserving homeomorphism. Let h : S^{1} → S^{1} be an orientation reversing homeomorphism. Prove that the rotation number of h^{−1} ∘ f ∘ h is equal to the negative of the rotation number of f mod 1,

ρ(h^{−1} ∘ f ∘ h) = −ρ(f) mod 1.

p. 94 Extra Problem 3.24. Define the map

f(x) = \{ \begin{matrix}a x + c  if 0 ≤ x ≤ c \\ a − a x  if c ≤ x ≤ 1, \end{matrix}

where c = \frac{1}{1 + a} and a^{2} = a + 1 are chosen so that f(c) = 1 using both definitions. Notice

that a = \frac{\sqrt{5} + 1}{2} > 1 is the golden mean.

(a) Draw the graph of f.

(b) Using the partition I_{1} = [0, c] and I_{2} = [c, 1] what is the transition matrix A for this partition.

(c) Prove that for any allowable sequence s ∈ Σ_{A}^{+}, there is a unique point h(s) such that f^{j}(h(s)) ∈ I_{s_{j}}.

Typeset by AMS-TEX

(d) Prove that σ_{A} on Σ_{A}^{+} is semi-conjugate to f on [0, 1]. (Any point which goes through c has two choices of symbols.)

(e) Prove that f is topologically transitive on [0, 1].

p. 213 Extra Problem 5.47. Let F_{a} : ℝ^{2} → ℝ^{2} be defined by the formula

f_{a}(x, y) = (1 − ax^{2} + y, x)

where a > 0 is a parameter.

(a) Final all the fixed points and points of period two.

(b) Find all values of the parameter a for which the points of period two are saddles.

p. 364 Extra problem on horseshoes (8.23\frac{1}{2}) Let f : ℝ^{2} → ℝ^{2} be a diffeomorphism with two hyperbolic saddle fixed points p_{1} and p_{2}. Assume there is a transverse heteroclinic point q_{1} in W^{u}(p_{1}) ∩ W^{s}(p_{2}) and another transverse heteroclinic point q_{2} in W^{u}(p_{2}) ∩ W^{s}(p_{1}). Discuss the correct form of the subshift of finite type which is conjugate to the hyperbolic invariant set which contains O(q_{1}) ∪ O(q_{2}) ∪ { p_{1}, p_{2} }.

p. 401 Extra Problem 9.33. Let S = {0} ∪ { 1/k^{p}: k is a positive integer }. Prove that dim_{b}(S) = 1/(p + 1) and dim_{H}(S) = 0.

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