ERRATA AND ADDITIONS FOR

DYNAMICAL SYSTEMS: STABILITY, SYMBOLIC DYNAMICS, AND CHAOS

by Clark Robinson

(*) Corrected in second printing. Also see the list of Corrections for the Second Edition of the book.

p. 20: (L. 8) “x ∈ I” should be “x ∈ I \ {p}” p. 22: (L. 5-8) This should read as follows: A direct calculation shows that for 2 < μ < 3 the

only fixed points of F_{μ}^{2} are those for F_{μ}, i.e., 0 and p_{μ}. (See Exercise 2.6.) Since F_{μ}^{2}(1/2) is above the diagonal, it follows that F_{μ}^{2}(x) is above the diagonal and x < F_{μ}^{2}(x) < p_{μ} for 1/2 ≤ x < p_{μ}. Therefore all the points in the interval [1/2, p_{μ}] converge to p_{μ} under interation by F_{μ}^{2}. Since |F′_{μ}(p_{μ})| < 1, it follows that all these points converge to p_{μ} under interation by F_{μ} as well.

p. 22: (L, 7) “below 1/2” should be “below p_{μ}” p. 23: (L. -18) “implies” should be “imply” p. 24: (L. 2) “d(f^{n}(x), ω(x))” should be “d(f^{n}(x), α(x))” p. 24: (L. 7) This should read as follows: Similarly, if f is invertible and y ∈ α(x) then α(y) ⊂ α(x)

and ω(y) ⊂ α(x). p. 24: (L. -9) “invariant” should be “positively invariant” p. 24: (L. -3) “S is closed” should be “S is a closed” p. 25: (L. 12) This should read as follows: …, although there are other … p. 28: (L. -9) remove “expansions have nonunique representations.” p. 32: (L. -4) “≤ λ|b − a|” should be “≥ λ|b − a|” p. 34: (L. 10) … derivative of f is always nonzero on the interval J, … p. 39: (L. 4) A stronger condition is … p. 39: (L. -9) … we describe such a point for p = 2. (An obvious change gives the general case.)

Let t be a … p. 39: (L. -3) Therefore d(σ^{n}(t), s) ≤ 3^{1−k}2^{−1}. p. 40 (L. 10) In the last section we showed p. 40 (L. -7) “onto” is not needed since it is part of the definition of semi-conjugacy. p. 42: (L. -8) change “2|z|.” to “2|z|,” p. 44: (L. 11) “h ∘ f(x) = g ∘ h” should be “h ∘ f(x) = g ∘ h(x)” p. 44: (L. -19) “and and” should be “and” p. 44: (L. -7)

= lim_{\begin{matrix}x→1 \\ x>1 \end{matrix}} h′_{0}(x).

Typeset by AMS-TEX

p. 44: (L. -4) Now let {x_{i}} be an arbitrary sequence with x_{i} ≠ 0 that … p. 44: (L. -1) “h_{0}” should be “h” p. 47: (L. 11) “[−2, 0] ∪ [1, 2].” should be “[−2, 0] ∪ [1, 2],” p. 51: (L. -6) “p ≥ F^{n}(0) < p + 1” should be “p ≤ F^{n}(0) < p + 1” p. 55: (L. 7) “F_{τ}^{n_{j}^{2}}(0) + m_{1}” should be “F_{τ}^{n_{j}^{2}}(0) + m_{2}” *p. 57: (2.6c) Prove for any point x with 0 < x < 1 that either (i) there is an integer k ≥ 0 such

that F_{μ}^{k}(x) = p_{μ} where p_{μ} is the fixed point or (ii) the ω(x) is the orbit of period 2. p. 58: (2.7d) … Then prove that W^{s}(O(q)) = (−p, 0) ∪ (0, p) \ O^{−}(0) where ±p are two of the

fixed points and O^{−}(0) is the backward orbit of the origin. p. 58: (2.8c) “O^{+}(p) = ω(p)” should be “O^{+}(p) ⊃ ω(p)” p. 58: (2.11) “Fμ^{′}(x) > 2” should be “|Fμ^{′}(x)| > 2” p. 58: (2.12) “|fμ^{′}(x)|ρ(f_{μ}(x))/ρ(x)” should be “|Fμ^{′}(x)|ρ(F_{μ}(x))/ρ(x)” p. 59: (2.16a) “I_{s_{0}...s_{n}0} to the left of I_{s_{0}...s_{n}1}” should be “I_{s_{0}...s_{n}1} to the left of I_{s_{0}...s_{n}2}” p. 60: (2.18) “|x_{j} − f^{j}(x)| < ϵ” should be “|x_{j} − F_{μ}^{j}(x)| < ϵ” p. 60: (21.6b) Remove the sentence “What effect does having a symbol 2 (where the slope is

negative on I_{2}) have on the order of the intervals?” p. 61: (2.25) … that are topologically conjugate by an orientation preserving homeomorphism.

… p. 61: “0 ≤ i < n” should be “0 ≤ i < p” p. 80: (L. 3) “slight” should be “slightly” p. 80: (L. 3) “slight” should be “slightly” p. 90: (3.13) It should read “Let Λ = ⋂_{n=0}^{∞} f^{−n}(I).” p. 101 (3.8d) I M(t) is a mtrix solution of the constant coefficient equation (**) with M(0) non-

singular, then p. 132: (L. -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. 133: (L. 26–27) These lines should read as follows:

B(v, w) = ∑_{ℓ=1}^{n}( ∑_{1≤i,j≤k} b_{i,j}^{ℓ}v_{i}w_{j})s^{ℓ}

= ∑_{1≤i,j≤k}(∑_{ℓ=1}^{n} b_{i,j}^{ℓ}s^{ℓ})v_{i}w_{j}.

p. 141 (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. 179: (Theorem 9.1) The region does not have to be simply connected. It should read: “either an

open subset of ℝ^{2} or D = S^{2}.”

*p. 193: (L. 8) “(λ − ϵα^{−1})^{n}2r” should be “(λ − ϵα^{−1})^{−n}2r” p. 195: (L. 6) See the separate sheet for an extended addition. p. 208: (5.40) It should read: “Assume that X̃_{1}(x, a) < 0, X̃_{1}(x, b) > 0, and X̃_{2}(x, a) = 0 = X̃_{2}(x, b)

for all x, ” p. 215: (L. 4) “th” should be “the” p. 215: (L. 9) “directions” should be “subspace” p. 215: (L. 10) “include” should be “includes”

p. 217: (L. -9) It should be “+2(\frac{∂^{2}f_{1}}{∂x_{1}∂x_{j}} (x_{0}, μ_{0}))(\frac{∂ϕ}{∂x_{1}} (a, μ_{0}))”

*p. 230: (L. -2) “F_{μ}(0, 0)(\frac{−F_{ϵϵ}(0, 0)}{F_{μ}(0, 0)})ϵ^{2}” should be “F_{μ}(0, 0)(\frac{−F_{ϵϵ}(0, 0)}{2 F_{μ}(0, 0)})ϵ^{2}”

*p. 232: (6.6) (a) … for 4A = 3(1 + B_{0})^{2}. (b) … at A = 3(1 + B_{0})^{2}/4 as A varies … p. 233: (6.10) The differential equation should read

ż = −\frac{8}{3} + xy

*p. 236: (L. 16) … is called an immersion provided the derivative of ϕ at each point is an isomorphism. The image of a one to one immersion is called an immersed submanifold. If an immersion is a homeomorphism then it is called an embedding and its image is called an embedded submanifold. (Some people require that an embedding is also proper, i.e., the inverse image of a compact set is compact.)

*p. 241: (L. 13) “T_{p}” should be “T_{p}M” p. 241: (L. 15) (iv) there exist 0 < λ < 1 and C ≥ 1 independent of p such that for all n ≥ 0 and

for all p ∈ Λ,

*p. 242: (L. 1) First sentence should be: Assume Λ is a hyperbolic invariant set for f with constants

0 < λ < 1 and C ≥ 1 giving the hyperbolic structure. p. 243: (L. -7, -9) These two line should read

= ⋃_{n≥0} f^{−n}(W_{ϵ}^{s}(f^{n}(p), f))

and

= ⋃_{n≥0} f^{n}(W_{ϵ}^{u}(f^{−n}(p), f)).

p. 248 (Line 8-10) In both the definitions of adjaceny matrix and transition matrix add the conditions that (ii) ∑_{j} a_{ij} ≥ 1 for all i, and (iii) ∑_{i} a_{ij} ≥ 1 for all j.

p. 256: (L. 10) It takes some argument that ⋂_{j=−n}^{n} F^{j}(S) has 4^{n} components. Any curve γ in S from x = −3 to x = 0 has an image F (γ) which reaches in S from x = −3 to x = 3. Similarly, any curve γ in S from x = 0 to x = 3 has an image F (γ) which reaches in S from x = −3 to x = 3. Applying this to the the top and bottom boundaries of the strips in F (S) ∩ S shows that ⋂_{j=0}^{2} F^{j}(S) has four strips all the way across S from x = −3 to x = 3. (There could be other components with just this argument. After proving the existence of invariant cones, it follows there are exactly four components.) By induction, ⋂_{j=0}^{n} F^{j}(S) has 2^{n} “horizontal” strips which reach all the way across S from x = −3 to x = 3. A similar argument applies to F^{−1} to show that ⋂_{j=−n}^{0} F^{j}(S) is made up of 2^{n} “vertical” strips which reach all the way across S from y = −3 to y = 3. A horizontal strip must intersect a vertical strip in at least one component. Therefore ⋂_{j=−n}^{n} F^{j}(S) has at least 4^{n} components. These are nested as n increases.

p. 256: (L. -11,-10) In the definition of the cones, “T_{p}M” should be “T_{p}ℝ^{2}” on both lines. p. 258: (L. -6) Compare with the arguement on the separate extended addition to page 195. p. 259: (L. -1) A horseshoe is any isolated invariant set which is (i) hyperbolic, (ii) zero topological

dimension, and (iii) topologically transitive. By using the existence of a Markov partition, Theorem IX.6.1, a diffeomorphism restricted to such an invariant set is conjugate to a subshift of finite type.

*p. 260: (L. 11) itinerary p. 262: (L. 4-10) For simplicity below, we take an adapted metric on Λ_{q}. (The adapted metric

implies that for x ∈ Λ_{q}, Df_{x} is an immediate contraction on 𝔼_{x}^{s} and an immediate expansion on 𝔼_{x}^{u}.) We extend the splitting 𝔼_{x}^{s} ⊕ 𝔼_{x}^{u} on Λ to a continuous (probably noninvariant) splitting 𝔼̂_{x}^{s} ⊕𝔼̂_{x}^{u} on a (perhaps smaller) neighborhood V of Λ_{q}. We use cones to show there is an invariant splitting which approximates 𝔼̂_{x}^{s} ⊕ 𝔼̂_{x}^{u} and extends the splitting 𝔼_{x}^{s} ⊕ 𝔼_{x}^{u} on Λ_{q}. For x ∈ V, using the adapted metric let

C^{s}(x) = {(ξ, η) ∈ 𝔼̂_{x}^{s} ⊕ 𝔼̂_{x}^{u}: |η| ≤ μ|ξ|}

and

C^{u}(x) = {(ξ, η) ∈ 𝔼̂_{x}^{s} ⊕ 𝔼̂_{x}^{u}: |ξ| ≤ μ|η|}

some 0 < μ < 1.

*p. 271: (L. 20) … so we want to measure Ĝ(z_{0}, ϵ) = H(z^{u}(z_{0}, ϵ))−H(z^{s}(z_{0}, ϵ)). Since Ĝ(z_{0}, 0) ≡ 0,

it is possible to write Ĝ(z_{0}, ϵ) = ϵG(z_{0}, ϵ). A zero of …. p. 298: (Step 2) Compare with the argument on the separate extended addition to page 195. p. 271: (L. 3b) Proof (of Theorem 4.6) We want to apply the Implicit Function Theorem to the

function G. We know that G(z_{0}, 0) = M(z_{0}) = 0 and

\frac{∂G}{∂v}(z_{0}, 0) = \frac{∂M}{∂v} (z_{0}) ≠ 0.

Let Σ be a transversal to the flow at z_{0} which contains the direction v. Applying the Implicit Function Theorem to G restricted to Σ, there is a function z^{∗}(ϵ) such that G(z^{∗}(ϵ), ϵ) ≡ 0, i.e., z^{∗}(ϵ) is a homoclinic point for small enough ϵ. The fact that the homoclinic point is transversal follows from the fact that the derivative is not zero.

p. 291 (L. 15) It should read

= ∏_{k=0}^{n} det(I − tf_{∗k})^{(−1)^{k+1}},

without the superscript j on f_{∗k}.

*p. 321: (L 11-12) “Ŵ^{σ}(O(p_{j})) ∩ Ŵ^{σ}(O(p_{j+1})) ≠ ∅” should be “Ŵ^{u}(O(p_{j})) ∩ Ŵ^{s}(O(p_{j+1})) ≠ ∅”

*p. 323: (Thm 12.3) “\frac{∂^{2}V}{∂x_{i}∂x_{j}} (x)” should be “(\frac{∂^{2}V}{∂x_{i}∂x_{j}} (x))”

p. 327 (7.13) Assume A is irreducible. *p. 328: (7.19) This refers to Example 4.1 on page 267, and not the one of page 253. *p. 328: (7.22) Let f_{A} be a hyperbolic toral automorphism on 𝕋^{n} with lift L_{A} to ℝ^{n}. Let g be a

small C^{1} perturbation of f_{A} with lift G to ℝ^{n}. Finally, let Ĝ = G − L_{A}. Let C_{b,per}^{0}(ℝ^{n}), C_{b,per}^{1}(ℝ^{n}), and Θ(Ĝ, v) be defined as in the proof of Theorem 5.1.

(a) Prove that Ĝ ∈ C_{b,per}^{1}(ℝ^{n}).
(b) Prove that Θ(Ĝ, ·) preserves C_{b,per}^{0}(ℝ^{n}).

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. 335: (Thm 1.2) The only thing that needs to be changed in the proof of Theorem 1.2 is to say “taking the limits in n and then η”. Note that ϵ ≤ η, so as η goes to zero, ϵ also must do to zero.

p. 344 (L. -9) This should read: “By letting ϵ go to zero, h_{sep}(K, f) = h_{span}(K, f).” p. 360 (Example 4.1:) The map should be

f(t, z) = (g(t), β z + \frac{1}{2}e^{2πti})

p. 364 (Ex. 8.28) It should read f(t, z) = (g(t), β z + \frac{1}{2} e^{2πti}).

(a) Prove for 0 < β < 1/(2\sqrt{2}), …

p. 368: (Theorem 1.1) … has a Liapunov function L such that (i) L is strictly decreasing off R(ϕ^{t}),

(ii) if L(x) = L(y) for x, y ∈ R(ϕ^{t}) then x ∼ y, and (iii) L(R(ϕ^{t})) is nowhere dense.

*p. 369: (L. 8) “Proposition 1.10” should be “Proposition 1.9” *p. 369: (L. 18-22) (b) The proofs that attracting sets and repelling sets are invariant are similar,

so we only look at A. Let x ∈ A and fix any real s. Then x ∈ ϕ^{t}(U) for all t ≥ 0, so ϕ^{s}(x) ∈ ϕ^{t+s}(U). Therefore, ϕ^{s}(x) ∈ ⋂_{t≥|s|} ϕ^{t+s}(U) = A. Since s is arbitrary, A is both positively and negatively invariant.

*p. 370: (L. 6) Since y ∉ U, y ∉ A. From the definition of Ω_{ϵ}^{+}, it follows that there is a T > 0 such

that ϕ^{T} (y) ∈ U so ϕ^{T} (y) ∉ A^{∗}. But Proposition 1.2 proves that …

*p. 370: (L. -8) Therefore, if x ∉ A^{∗}, then V (ϕ^{t}(x)) goes to zero p. 372: (Theorem 1.8) … a Liapunov function L : M → ℝ such that (i) L is strictly decreasing off

P, (ii) if L(x) = L(y) for x, y ∈ P then x ∼ y, and (iii) L(P) is nowhere dense. p. 372: (L. 24) If x, y ∈ P and x ≁ y then there is a k such that one of the points is in A_{k} and the

other is in A_{k}^{∗}, so L_{k}(x) ≠ L_{k}(y) and L(x) ≠ L(y). p. 380: (L. -4) “≤ δ + ϵ” should be “≤ ν + ϵ”. p. 385: (L. 8) there is a p′ ∈ H_{p} and q′ ∈ H_{q} such that … p. 398: (Theorem 7.1) This should read: … f : M → M is an Anosov diffeomorphism (f has a

hyperbolic structure on all of M). p. 400: (L. -5) … and so is a conjugacy. p. 402: (L. 3) is an stable disk at x p. 402: (L. 6–9) By construction of the disks and definition of F^{t},

(ϕ^{t}(x), ψ^{τ(t,x,h(x))}(h(x))) = F^{t}(x, h(x)) ∈ D^{u}(ϕ^{t}(x), η) ∩ D^{s}(ϕ^{t}(x), η),

but also

(ϕ^{t}(x), h ∘ ϕ^{t}(x)) ∈ D^{u}(ϕ^{t}(x), η) ∩ D^{s}(ϕ^{t}(x), η),

so by uniqueness of this point

h ∘ ϕ^{t}(x) = ψ^{τ(t,x,h(x))}(h(x)).

p. 402: (Proposition 8.2) A flow ϕ^{t} is called flow expansive on Λ provided given ϵ > 0, there exists a δ > 0 such that if x_{1}, x_{2} ∈ Λ with

d(ϕ^{t}(x_{1}), ϕ^{σ(t)}(x_{2})) < δ

for all t where

lim_{t→±∞} σ(t) = ±∞,

then x_{2} = ϕ^{s}(x_{1}) for some |s| ≤ ϵ. p. 407: (L. 5) Exercise 9.38 should be Exercise 9.37. p. 408: (L. 5)

“d(D^{−i}(x_{j}), x_{j−i}) ≤ δ(\frac{1}{q} + ⋯ + \frac{1}{q^{i}} ) ≤ δ”

should be

“d(D^{−i}(x_{j}), x_{j−i}) ≤ δ(\frac{1}{2} + ⋯ + \frac{1}{2^{i}} ) ≤ δ.”

p. 409: (Problem 22) It is necessary to assume that all points are nonwandering. (Problem 22b)

Prove that W^{s}(q) is dense in M. p. 427: (L. 2-3) These should be “−f̂(x)D(β_{r})_{x}” not “+f̂(x)D(β_{r})_{x}” p. 427: (L. -3) … the transversality condition with respect to R(f). p. 428 (L. 3-5) Assume f is C^{1} structurally stable. Then by Theorem 4.3 (or Exercise 10.11(c)),

it then follows that f also satisfies the transversality condition with respect to R(f). p. 434: (L. -7) … complete spaces (closed balls in normed linear spaces) since … p. 438: (L. -3)

∡(C^{u}) = sup\{\frac{|w − w′|}{|v|}: (v, w), (v, w′) ∈ C^{u} ⊂ T_{x}X × Y \}.

p. 439: (L. -9) Compare with the argument on the separate extended addition to page 195. p. 443: (L. 18) “Since the invariant section has 𝔼_{x}^{u} as a graph, the unstable bundle is C^{1}.” should

be “By uniqueness, the invariant section has 𝔼_{x}^{u} as a graph and the unstable bundle is C^{1}.”

The following people were some of those who told me about errata in the book: Youngna Choi, James Jacklitch, Ming-Chia Li, Jody Sorensen, and Dick Swanson.

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

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