Define the sequence as follows: and for . What can we say about its behavior as ?
The logistic map leaves the interval [0,1] invariant (as a set), so for all . There are two fixed points: 0 and 3/4.
Can ever be 0? If is the first index this happens, then must be . Working backwards, we find , and . But this is impossible since all elements of the sequence are rational. Similarly, if is the first index when , then and , a contradiction again. Thus, the sequence never stabilizes.
If had a limit, it would have to be one of the two fixed points. But both are repelling: , so and . This means that a small nonzero distance to a fixed point will increase under iteration. The only way to converge to a repelling fixed point is to hit it directly, but we already know this does not happen. So the sequence does not converge.
But we can still consider its upper and lower limits. Let’s try to estimate from below. Since for , the sequence increases as long as . Since we know it doesn’t have a limit, it must eventually break this pattern, and therefore exceed 3/4. Thus, .
This can be improved. The second iterate satisfies for between and . So, once (which, by above, happens infinitely often), the subsequence increases until it reaches . Hence .
The bound is best possible if the only information about is that the sequence does not converge. Indeed, is a periodic point of , with the corresponding iteration sequence .
Further improvement is possible if we recall that our sequence is rational and hence cannot hit exactly. By doubling the number of iterations (so that the iterate also fixes but also has positive derivative there) we arrive at the fourth iterate . Then for , where is the next root of after , approximately . Hence .
This is a colorful illustration of the estimation process (made with Sage): we are covering the line with the iterates of , so that each subsequent one rises above the line the moment the previous one falls. This improves the lower bound on from 0.75 to 0.9 to 0.92.
Although this process can be continued, the gains diminish so rapidly that it seems unlikely one can get to 1 in this way. In fact, one cannot because we are not using any properties of other than “the sequence is not periodic.” And it’s not true that for every non-periodic orbit of . Let’s return to this later.