Question 104
Let $a$ and $b$ be real numbers. Define $f_{a,b}: \mathbb{R}^2 \to \mathbb{R}^2$ by $f_{a,b}(x, y) = (a - b y - x^2, x)$. If $P = (x, y) \in \mathbb{R}^2$, define $f_{a,b}^0(P) = P$ and $f_{a,b}^{k+1}(P) = f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$.
The set $\operatorname{per}(a, b)$ of the *periodic points* of $f_{a,b}$ is the set of points $P \in \mathbb{R}^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$.
Fix $b$. Prove that the set $A_b = \{a \in \mathbb{R} \mid \operatorname{per}(a, b) \neq \emptyset\}$ admits a minimum. Find this minimum.
The set $\operatorname{per}(a, b)$ of the *periodic points* of $f_{a,b}$ is the set of points $P \in \mathbb{R}^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$.
Fix $b$. Prove that the set $A_b = \{a \in \mathbb{R} \mid \operatorname{per}(a, b) \neq \emptyset\}$ admits a minimum. Find this minimum.