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 $$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3} \Big(r_{i+1}-\frac{r_i^2}2\Big) $$ for $i=0,1,\dots$. Questions:

- Is it true that the only periodic sequence $(r_i)$ here is the constant one with $r_i=1$ for all $i$?
- 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 $i$?
- 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 $$\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}\right)\ge \sqrt[4]{\dfrac{a^4+b^4+c^4+d^4}{4}}$$ 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 $\{(r_0,r_1)\in D\colon0<r_0<3,r_4<r_0\}$ (red, in both pictures), $\{(r_0,r_1)\in D\colon0<r_0<3,r_5<r_1\}$ (gray), and $\{(r_0,r_1)\in D\colon0<r_0<3,r_0r_1r_2r_3<1\}$ (green); the horizontal axes here are for $r_0$ and the vertical ones for $r_1$.

Khue: the recurrence equation in my answer is equivalent, up to notation, to equation $$2\xi_{k+1}-\xi_k^2=\Big(2\xi_{k+2} - \xi_{k+1}^2\Big)\xi_{k+1}^3$$ in the answer byjohannesvalksat math.stackexchange.com/questions/852070/… concerning the inequality for $a,b,c,d$. $\endgroup$