Errata

• Theorem 2.1 contains a typo. The correct statement is

  If \(a_n \leq b_n \leq c_n\), for all \(n\) and

  \(\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} b_n = L\)

  then

  \(\lim_{n \rightarrow \infty} c_n = L\).

• Theorem 2.1 uses the subscript\(a_b\), which should be \(b_n\).
• Chapter 2, Application 20 should state that the minimum terms desired are \(n = 0,1\) and \(2\).
• Chapter 2, Exploration 46 is missing a condition. The set in item 4 of this exploration should be \(S = \{x \in \mathbb{R} | x = 1/n, n \in \mathbb{Z} \setminus \{0\}\}\)
• Chapter 2, Exploration 58 indicates the dots are green and red. The dots on the website have been changed to accomodate color-blind students. The color relationship is described in each exploration.
• Chapter 2, Conjecture 60 should refer to the sequence of supremums \(\{v_N\}\).
• The second bullet of Theorem 4.6 should read \(|F_{c_0}'(\gamma(c))|\neq 1\).
• Chapter 6, Explorations 2-8 should be Applications 2-8.
• Definition 7.2 contains a typo. The correct version is below.

  Let \(X\) be a metric space with metric \(d\) and suppose \(U,\, V \subset X.\) A set \(U\) is dense in a set \(V\) if for every \(\epsilon>0\) and every point \(v \in V\), there exists a \(u \in U\) such that \(d(u,v)< \epsilon\).

• Section 8.4 - These exercises as written are problematic. There is not a homeomorphism between \(\Sigma_2\) and \([0,1]\) and thus creating this topological conjugacy cannot be done. For example, the sequences \(.1\) and \(.0\bar{1}\) are distinct in \(\Sigma_2\) but as binary sequences they both represent \(x=1/2.) It turns out that (\Sigma_2\) is homeomorphic to the Cantor Set. The authors suggest that you omit proofs 23-29 and replace them with the following (possibly to be done at the end of section 8.2). Prove that the doubling map (D\) is chaotic on $\([0,1]\)? Remember that the metric \(D\) on \(\Sigma_2\) is not the metric on \(\R\) and you will need to adjust accordingly.