ERRATA AND ADDITIONS FOR THE 2008 PRINTING OF THE SECOND

EDITION OF

DYNAMICAL SYSTEMS: STABILITY, SYMBOLIC DYNAMICS, AND CHAOS

CLARK ROBINSON

p. 33: (Section 2.4.3) Explanation: S. Zeller and M. Thaler (“Almost sure excape from the unit interval under the logistic map”, Amer. Math. Monthly 108 (2001), pages 155 – 158.) have a simpler proof based on the earlier thesis of S. Zeller (“Chaosbegriffe der topologishen Dynamik”, Dipolmarbeit, Salzburg, 1991). The map

y = ϕ(x) = \frac{2}{π} arcsin \sqrt{x}

is a conjugacy between F_{4}(x) and g_{4}(y) = 1 − |1 − 2y|, g_{4}(y) = ϕ ∘ F_{4} ∘ ϕ^{−1}(y). For μ > 4, F_{μ}([0, 1]) = [0, \frac{μ}{4}], so it is natural to scale ϕ by the factor \frac{μ}{4} to investigate F_{μ}. Let ϕ_{μ}(x) = \frac{μ}{4} ϕ(\frac{4}{μ}x). Define the map g_{μ} by

g_{μ}(y) = ϕ_{μ} ∘ F_{μ} ∘ ϕ_{μ}^{−1}(y).

Then a simple calculation shows that |g′_{μ}(y)| ≥ \sqrt{μ} for all y ∈ [0, ϕ_{μ}(a)], so F_{μ} has can invariant Cantor set. The proof is a follows. Let y = ϕ_{μ}(x) for x ∈ [0, 1). Because ϕ is a conjugacy of F_{4} and g_{4},

ϕ′(F_{4}(x)) F′_{4}(x) = (2 sign(1 − 2x)) ϕ′(x).

Also, F′_{μ}(x) = \frac{a}{4}F′_{4}(x). Thus

g′_{4}(y) = \frac{ϕ′_{μ}(F_{4}(x)) F′_{μ}(x)}{ϕ′_{μ}(x)} = \frac{a ϕ′(F_{4}(x)) F′_{4}(x)}{4 ϕ′_{μ}(x)}

= sign(1 − 2x)) \frac{μ ϕ′(x)}{2 ϕ′_{μ}(x)}

= sign(1 − 2x)) \sqrt{μ} (\frac{1 − \frac{4}{μ}}{1 − x} )^{\frac{1}{2}} .

Since \frac{4}{μ} < 1, the last term is greater than \sqrt{μ}. p. 38: (Section 2.5) In this section, we show that the dynamics of F_{μ} on Λ can be understood

in terms of a map on a symbol space made up by points which are sequences of 1’s and 2’s. The map on the symbol space is said to give the symbolic dynamics for the map. At least in a theoretical way, we can determine the periodic points. We also want to show that there are points whose orbit is dense in the cantor set Λ and points with other complicated dynamics. By introducing symbols to describe the location of a point, the dynamics of a point in the Cantor set can be determined by means of a sequence of these symbols. Because many different patterns of symbols can be written down, points with many different types of dynamics can be shown to exist.

p. 50: (L. -23) Explanation: The covering space ℝ of S^{1} can be thought of as measuring the angle without reducing modulo 2π, or modulo 1, in the coordinates on ℝ. Thus, the points t, t+1, and t+2 in ℝ all represent the same point in §^{1}. In the same way, the lift of f : S^{1} → S^{1} to F : ℝ → ℝ gives the new location without reducing modulo 1. The difference F (t) − t is the amount the point is move around the circle without reducing modulo 1. p. 50: (L. -9) Explanation: Let F_{λ}(t) = t + λ be the rigid rotation. Then the change of angle,

F_{λ}(t)(t) − t = λ is the same for any point. For an arbitrary homeomorphism of S^{1}, the change F (t) − t can vary with the point t. The quantity

F^{n}(t) − t = [F^{n}(t) − F^{n−1}(t)] + [F^{n−1}(t) − F^{n−2}(t)] + ⋯ + [F (t) − t]

is the total change of angle by the n^{th}-iterate without reducing modulo 1. The average change of angle for one iterate by the first n-iterates is

\frac{1}{n}\{F^{n}(t) − t\} = \frac{1}{n}\{[F^{n}(t) − F^{n−1}(t)] + [F^{n−1}(t) − F^{n−2}(t)] + ⋯ + [F (t) − t]\}.

Taking the limit as n goes to infinity, lim_{n→∞} \frac{1}{n}\{F^{n}(t)−t\} gives the average change of angle for one iterate along the whole orbit. This last limit is used to define the rotation number of the map on the circle.

p. 79: (L. -8) there is an allowable word w such that p. 96: (L 5) Explanation: The norm of a matrix can be calculated in terms of an eigenvalue

of a related matrix. Notice that

|Ax|^{2} = (Ax)^{t}Ax = x^{t}A^{t}Ax.

The maximum of this quantity as x varies over unit vectors is the square of the norm of A. The matrix A^{t}A is symmetric and so has real eigenvalues. If λ_{1} is the largest eigenvalue with unit eigenvector v^{1} then

v_{1}^{t}A^{t}Av_{1} = v_{1}^{t}λ_{1}v^{1} = λ_{1}.

Therefore the norm of A is the square root of the largest eigenvalue of A^{t}A, ∥A∥ = \sqrt{λ_{1}}. p. 114: (L. 6) v ∈ V^{u} should be v ∈ V^{c}: “as t → ±∞, so v ∈ V^{c}.” p. 134: (Line -7 to -4) Replace with: “If U is a region where f(x) is defined and C^{1} and

V ⊂ U is a compact subset, then we can let K = sup{∥Df_{x}∥ : x ∈ V }. By the Mean Value Theorem,

|f(x − f(y)| ≤ K|x − y|

if the line segment from x to y is contained in V.” p. 143: (Line 7–9) For x_{0} ∈ U take b > 0 such that the closed ball B̄(x_{0}, b) ≡ {x : |x − x_{0}| ≤

b} ⊂ U. The function f is Lipschitz … for all x, y ∈ B̄(x_{0}, b). p. 389: (Line 6) The way to calculate the limits of the wedge product is to start with an

orthonormal basis {v^{0,1}, …, v^{0,m}} of tangent vectors at x^{0} = x. Let x^{k} = f^{k}(x). Assume by induction that we have defined an orthonormal basis {v^{k−1,1}, …, v^{k−1,m}} at x^{k−1}. Applying the derivative at x^{k−1}, let w^{k,j} = Df_{x^{k−1}}v^{k−1,j} be the image vectors. Apply the Gram-Schmidt process to construct a basis of perpendicular vectors:

z^{k,m} = w^{k,m}
z^{k,m−1} = w^{k,m−1} − \frac{w^{k,m−1} · z^{k,m}}{|z^{k,m}|^{2}} z^{k,m}
z^{k,j} = w^{k,j} − ∑_{i=j+1}^{m} \frac{w^{k,j} · z^{k,i}}{|z^{k,i}|^{2}} z^{k,i}  for 1 ≤ j ≤ m − 1.

We get an orthonormal basis of vectors at x^{k} by letting

v^{k,j} = \frac{z^{k,j}}{|z^{k,j}|}.

This completes the induction process. The multiplicative factor of the j^{th}-vector is

r_{j}^{(k)} = |w^{1,j}| ⋯ |w^{k,j}|.

The volume of the parallelograms spanned by {z^{k,m−j+1}, …, z^{k,m}} is the same as that spanned by the {w^{k,m−j+1}, …, w^{k,m}}, which is r_{m−j+1}^{(k)} ⋯ r_{m}^{(k)}. Thus the growth rate of this volume as k goes to infinity is

λ_{m−j+1} + ⋯ λ_{m} = lim_{k→∞} \frac{1}{k} log(r_{m−j+1}^{(k)} ⋯ r_{m}^{(k)})

= lim_{k→∞} \frac{1}{k} log(r_{m−j+1}^{(k)}) + ⋯ lim_{k→∞} \frac{1}{k} log(r_{m}^{(k)}),

and

λ_{m−j+1} = lim_{k→∞} \frac{1}{k} log(r_{m−j+1}^{(k)})

= lim_{k→∞} \frac{1}{k} ∑_{i=1}^{k} log(|w^{i,m−j+1}|).

p. 421: In the proof of Theorem 5.4, if we assume that ℝ(f) is hyperbolic, then it is possible to take the chain componenets rather than the sets cl(H_{p}) in the decomposition.

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Send comments about the book or further errata to the author at clark (at) math.northwestern.edu