Are the claims about repeating states and space in Netflix's "A Trip to Infinity"'s based on real research? I just watched Netflix's documentary on infinity "A trip to infinity". They have an example where you put an apple in a perfectly sealed box. They make a claim that seems odd to me.
The apple will decay etc etc but because there are a finite number of possible states for the contents in the box, it must eventually reach a state it was in before (all good up to here) and so it will eventually become an apple again.
Why must it eventually become an apple again? It will enter a cycle but there's no need for that cycle to contain the initial state. Wouldn't the cycle would very likely be a cycle through a bunch of states of very high entropy?
Similarly they claim that if the universe is infinite, then our region of space must be repeated an finite number of times elsewhere. Why? Does something force a repeat of our region? Sure, some region might repeat but it doesn't have to be an interesting region.
Are there papers/theorems behind these claims that forces repeats to be "interesting"?
They have some serious scientists in the documentary, so it would be weird for this to be completely wrong.
 A: As far as I can tell, the apple claim refers to the ergodic hypothesis, which implies that each microstate of a system will be assumed over time. In the apple example, the macrostate is the box containing a given collection of atoms with a given energy. The apple being whole is one microstate and thus it will need to be assumed again by the ergodic hypothesis.
Now, the ergodic hypothesis is, well, only a hypothesis. It is impossible to empirically check as such, but we use it as the foundation of statistical mechanics, from which several empirically confirmed statements are derived. Also, all known toy systems that break the ergodic hypothesis are somewhat pathological and I see no reason to assume that the apple system is one of them.
If we assume true quantum randomness, we can see how the apple might become whole again: It is possible to find a chain of state transitions from every given state to the apple being whole: Atoms just need to collide with each other in a given way. This chain is admittedly very unlikely to happen, but the probability is not total zero. Thus if you wait long enough, it will eventually happen, though you cannot say when. (Mind that this doesn’t prove ergodicity as that would require showing that all states are equally likely.)
You might compare this to applying random transformations to a Rubik’s cube: Most of the time, you will get an unsolved cube, but there clearly is a chain of transformations that solves the cube from any given state¹. Thus if you apply random transformations long enough, you will obtain the solved cube. Also, all possible configurations of the cube are clearly equally likely, including the solved one. Mind that if you have a deterministic chain of transformations, it may not ever visit the solution, but a random one is bound to.
¹ I here assume that the cube has not been tampered with, e.g., by prying out an edge stone and flipping it.

Wouldn't the cycle would very likely be a cycle through a bunch of states of very high entropy?

The thing about entropy maximisation is that it is only a statistical statement, i.e., it only holds when looking at large ensembles of particles (or similar) and only with a probability that is sufficiently close to one to assume it to be true for all practical purposes.
The apple example destroys the underlying assumptions by waiting an infinite amount of time.
A: That is a bogus claim.
If we imagine that the apple rots out to its constituent atoms, and those atoms form a gas, there will be instants in time where a few of the hydrogen atoms come close to an oxygen and a few carbons all at the same spot, and the result of that fluke is that for a microsecond, the transient agglomeration of those atoms vaguely resembles a sugar molecule- before those atoms go their own way. Then you wait a year before it happens again.
(And for that apple to reconstitute itself, the constituent molecules have to be put together out of atoms, which requires energy. With no energy input to the box, you can't build molecules.)
Now notice that in an apple there are about 10^23 atoms, and that there is only one right way to put all of them together again into a duplicate of the original apple. There are, on the other hand, a truly huge (but finite!) number of wrong ways to put those atoms together, none of which even vaguely resembles an apple. How huge?
Pick an atom, one out of 10^23. Pick another at random; your chance of getting the next one right is one out of (10^23 - 1). The chance of getting the next one right is one out of (10^23 - 2), and so on until you have chosen every atom correctly to exactly reconstitute every molecule in the original apple. The cumulative probability is going to look  like one out of  (10^23) x (10^23 - 1) x (10^23 - 2) x (10^23 - 3) x ..., repeated 10^23 times.
Just the first four terms give you a probability of one chance in 10^102, for comparison there are only about 10^80 fundamental particles in the entire universe.
This means that reconstituting the same apple out of randomly picked atoms will essentially never never never never never happen- especially if you do not provide the energy input necessary to build the molecules. So the claim is bogus.
A: I have not seen the referenced documentary but seemingly 'absurd' claims like you have mentioned can be made because infinity is absurdly large.
If we do not care whether it is allowed by the laws of physics or not, every arrangement of a group of atoms has to be repeated infinite number of times in a collection of infinite number of atoms.# So if the universe is infinite, every place - which is a particular arrangement of atoms - will repeat infinite number of times. Leave just one such configuration, you fall short of infinity.
Similarly, if we wait for infinite time and the atoms in the apple keep on rearranging randomly, we will get every arrangement of atoms - including the arrangement of atoms in the original state of the apple - infinite number of times.
I think such claims are made just to emphasize the largeness of infinity. They consider just the statistics, the physics may have something else to say.
EDIT:
# Perhaps this sentence is not conveying what I intend to say so I am trying to rephrase/explain it in brief.
To make a collection of atoms infinite we have to add every possible combination of atoms into it. If we leave any such combination, we fall short of infinity. Not only this, every combination of atoms has to be repeated till it can not be repeated any further - if there really comes such a point after which the combination can not be repeated further. This means, we have to add every combination of atoms 'infinite' number of times. [Use of 'infinite' to describe infinity may make the concept cyclic but here it is to convey just the idea mentioned before.]
Of course, we are talking about absolute infinity which is just an idea developed by mathematicians. Physically, we can not achieve infinity. Number of atoms in the universe is not infinite and physics imposes restrictions on how atoms can be arranged. But when we are discussing just the 'idea', we need to consider every possible combination of atoms 'infinite' number of times to achieve the infinity.
