# Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence

for $i=0,1,\dots$. Questions:

1. Is it true that the only periodic sequence $(r_i)$ here is the constant one with $r_i=1$ for all $i$?
2. Take any natural $n$. Suppose that the sequence $(r_i)$ is periodic with period $n$ and $r_0\cdots r_{n-1}=1$. Does it then always follow that $r_i=1$ for all $i$?
3. Suppose that the sequence $(r_i)$ is periodic with period $4$ and $r_0\cdots r_3=1$. Does it then always follow that $r_i=1$ for all $i$?

Of course, Question 2 is a weaker version of Question 1, and Question 3 is a weaker version of Question 2. If the answer to Question 3 is yes, that would answer affirmatively the question on the inequality

for $a,b,c,d>0$, posed at MO; cf. mathSE.

Questions 1–3 are illustrated in the (rather suggestive) pictures below, showing the sets $\{(r_0,r_1)\in D\colon0 (red, in both pictures), $\{(r_0,r_1)\in D\colon0 (gray), and $\{(r_0,r_1)\in D\colon0 (green); the horizontal axes here are for $r_0$ and the vertical ones for $r_1$.