ADDITIONAL PROBLEMS FOR

DYNAMICAL SYSTEMS: STABILITY, SYMBOLIC DYNAMICS, AND CHAOS

by Clark Robinson

Also see the list of Extra Problems for the Second Edition of the book.

p. 202 Additions to problem. 5.4.

(b) Assume that g is Lipschitz. Prove that σ is a Lipschitz function.

(c) Assume that Λ is an open subset of a Euclidean space ℝ^{k}, Y is a closed ball in ℝ^{n}, and g is a C^{r} function on Λ × int(Y). Apply the Implicit Function Theorem to G(x, y) = g(x, y) − y to prove that σ is a C^{r} function.

p. 328 Extra problem for Anosov Diffeomorphisms: Let f_{A} be a hyperbolic toral automorphism on 𝕋^{n} with lift L_{A} to ℝ^{n}. Let g be a map of 𝕋^{n} which is homotopic to f_{A}. Use Problem 5.30 to prove that g is semi-conjugate to f_{A}.

p. 329 Improved wording for 7.25. (A horseshoe as a subsystems of a hyperbolic toral automorphism.) Let f_{A_{2}} : 𝕋^{2} → 𝕋^{2} be the diffeomorphism induced by the matrix

A_{2} = ( \begin{matrix}2 & 1 \\ 1 & 1 \end{matrix} )

discussed in Example 5.4. Let R_{1a} be the rectangle used in the Markov partition for this diffeomorphism. Let g = f_{A_{2}}^{2} and

Λ = ⋂_{j=−∞}^{∞} g^{j}(R_{1a}).

Prove that g : Λ → Λ is topologically conjugate to the two-sided full two-shift σ : Σ_{2} → Σ_{2}. Hint: R_{1a} plays the role that S played in the construction of the geometric horseshoe. Prove that g(R_{1a}) ∩ R_{1a} is made up of two disjoint rectangles. (These rectangles are similar to V_{1} and V_{2} in the geometric horseshoe.) Looking at the transition matrix for the Markov partition for f_{A_{2}} may help.

p. 332 Extra problem. 7.47. Assume f : ℝ^{2} → ℝ^{2} is a diffeomorphism with a compact invariant set Λ. Assume Λ satisfies all the conditions for a hyperbolic structure except possibly the continuity of the splitting. Assume the dimension of each of the subspaces in the splitting has dimension one. Prove that the splitting is continuous. Hint: Consider the argument in the separate addition to page 195.

p. 332 Extra problem. 7.48. Assume f : ℝ^{2} → ℝ^{2} is a diffeomorphism with a compact hyperbolic invariant set Λ. Prove that if V is a small enough neighborhood of Λ, then the maximal invariant set in V,

Λ_{V} ≡ ⋂_{n∈ℤ} f^{n}(V),

Typeset by AMS-TEX

has a hyperbolic structure. Hint: See the proof of Theorem 4.5.

p. 332 Extra problem. 7.49. Let f : ℝ → ℝ be C^{1} function Assume that f(0) = 0 and f′(0) > 1. Assume that q ≠ 0 is a point such that (i) f(q) = 0, (ii) there is a choice of a backward orbit {q_{k}}_{k≤0} with q_{0} = q, f(q_{k−1}) = q_{k} for k ≤ 0, f′(q_{k}) ≠ 0 for k ≤ 0, and q_{k} converges to 0 as k goes to −∞, i.e., q ∈ W^{u}(0). Prove that there is an invariant set containing q and 0 which is conjugate to a one-sided subshift of finite type. Hint: Since f is not a diffeomrophism, Theorem 4.5 does not apply. Find intervals which work correctly as the boxes do in the proof of Theorem 4.5.

p. 332 Extra problem. 7.50. Assume the flow ϕ^{t} is gradient-like with Liapunov function L. a. Prove that Ω(ϕ^{t}) = Fix(ϕ^{t}). b. Assume that L(Fix(ϕ^{t})) is a totally diconnected subset of ℝ. Prove that R((ϕ^{t}) =

Fix(ϕ^{t}).

p. 332 Extra problem. 7.51. Consider the map given in polar coordinates by

F ( \begin{matrix}θ \\ r \end{matrix} ) = ( \begin{matrix}8 θ − \frac{π}{8} \\ 2 + \frac{1}{16} r + \frac{1}{16} sin(θ) \end{matrix} )

for 0 ≤ θ ≤ \frac{π}{2} and 1 ≤ r ≤ 3. (Notice this definition is only for part of the plane.) Let

V_{L} = {(θ, r): 0 ≤ θ ≤ 3 π/32, 1 ≤ r ≤ 3}
V_{R} = {(θ, r): π/4 ≤ θ ≤ 11 π/32, 1 ≤ r ≤ 3}.

Prove that the maximal invariant set in V_{L} ∪ V_{R} is hyperbolic.

For information about the book contact CRC Press, Inc 2000 Corporate Blvd., N.W. Boca Raton, Florida 33431-9868 800-272-7737