Dynamics of an inequality

The dynamics D(ri,ri+1)(ri+1,ri+2)D on the set D:={(x,y)R2:x>0,y>x2/2} is given by the recurrence
ri+2=r2i+12+1r3i+1(ri+1r2i2)
for i=0,1,. Questions:

  1. Is it true that the only periodic sequence (ri) here is the constant one with ri=1 for all i?
  2. Take any natural n. Suppose that the sequence (ri) is periodic with period n and r0rn1=1. Does it then always follow that ri=1 for all i?
  3. Suppose that the sequence (ri) is periodic with period 4 and r0r3=1. Does it then always follow that ri=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
14(a2b+b2c+c2d+d2a)4a4+b4+c4+d44
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 {(r0,r1)D:0<r0<3,r4<r0} (red, in both pictures), {(r0,r1)D:0<r0<3,r5<r1} (gray), and {(r0,r1)D:0<r0<3,r0r1r2r3<1} (green); the horizontal axes here are for r0 and the vertical ones for r1.
enter image description here

Answer

Attribution
Source : Link , Question Author : Iosif Pinelis , Answer Author : Community

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