Do write down the commutative diagram, as I think it helps keep things straight b. *Conjugacy* (You can simply state the extra assumption necessary to extend the definition of semi-conjugacy) c. *Criterion for a attractive orbit* (i.e. Theorem 2.25 in the "Critical orbits" section). d. *Sensitive dependence on initial conditions* (Essentially, [Claim 2.44](https://www.marksmath.org/classes/Fall2021ChaosAndFractals/chaos_and_fractals_2021/subsection-chaos.html#claim-sensitive_dependence)) but applied to an arbitrary function $f$, rather than the doubling map $d$. 2. **Apply conjugation**: Let $f(x) = x^2$ and let $\varphi(x) = 2x+1$. a. Conjugate $f$ by $\varphi$. b. Use your conjugation to describe the critical orbit of $g$. c. Given your knowledge of $f$, what can you say about the dynamics of $g$? 3. **The doubling map**: Let $d(x) = 2x \, \% \, 1$ denote the doubling map that maps $H:[0,1)$ to itself. a. Find a point $x_0\in H$ that has period 5 expressed as a binary expansion. b. Use the geometric series formula to express $x_0$ as a fraction. 4. **Applying the doubling map**: Continuing on with the previous problem, suppose I tell you that the doubling map is semi-conjugate to $f(x)= \frac{x^2}{2}+3 x-\frac{5}{2}$ via the semi-conjugacy $\varphi(x) = 4 \cos (2 \pi x)-3$, i.e. $f\circ\varphi = \varphi\circ d$. a. Find a point of period 5 for $f$. b. What does your knowledge of $d$ tell you about the general behavior of $f$? 5. **Finding a super-attractive parameter**: Consider the family $f_c(x) = x^2 - 4x + c$. Write down an equation that $c$ must satisfy to ensure that $f_c$ has a super-attractive orbit of period 3.