Errata and Additions for Dynamical Systems: Stability, Symbolic Dynamics, and Chaos

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 2 μ are those for Fμ, i.e., 0 and pμ. (See Exercise 2.6.) Since F 2 μ(1/2) is above the diagonal, it follows that F 2 μ(x) is above the diagonal and x < F 2 μ(x) < pμ for 1/2 ≤ x < pμ. Therefore all the points in the interval [1/2, pμ] converge to pμ under interation by F 2 μ . Since |F ′ μ(pμ)| < 1, it follows that all these points converge to pμ under interation by Fμ as well. p. 22: (L, 7) “below 1/2” should be “below pμ” p. 23: (L. -18) “implies” should be “imply” p. 24: (L. 2) “d(f(x), ω(x))” should be “d(f(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(t), s) ≤ 32. 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 x→1 x>1 h0(x).