I am studying this application of Poincaré Recurrence Theorem and I have some questions.
1 - it states that If $S$ is a bounded space with measure $µ$ and $T : S \to S$ is a measure-preserving transformation, then for any set with positive measure $B \subseteq S$, the subset $A \subseteq B$ of points that never recur to $B$ has measure zero.
Does $S$ must be bounded? All the other references that I've seen does not require it. Why does the author require $S$ to be bounded?
2 - Page 12: The Poincaré recurrence theorem states that the set of points in $B$ that will never return to $B$ no matter how many time intervals are observed has measure zero. This means that with probability one, the system’s initial configuration is not in this subset, and eventually will return to $B$.
I don't get this sentence. How can we guarantee that the poll balls are not in this mesuare zero subset? And then the system's initial configuration will return to $B$?
Again, on page 13, Finally, although the recurrence theorem guarantees recurrence with probability one, it does not guarantee that the system recurs for every single point in B. In fact, there may be infinitely many points that do not return to B. However, the theorem does assert that the set of points in B that do not return to B has measure zero – even though B has positive measure. Thus, although it is possible that the billiard table’s initial configuration is one that never recurs, it is a zero probability event; consequently, the probability of recurrence is one.
I don't get this, is it saying that the it is possible that the billiard table's initial configuration can never recurs or it will always recur?
thanks in advance.
Answering directly to your questions:
$1$. Of course that $S$ need not be bounded.
$2$. "This means that with probability one, the system’s initial configuration is not in this subset, and eventually will return to $B$."
is intended to mean
"This means that with probability one, the system’s initial configuration is not in this subset, and so it is with probability one in the complement of that subset, which means that it eventually will return to $B$."
$3$. "Thus, although it is possible that the billiard table’s initial configuration is one that never recurs, it is a zero probability event; consequently, the probability of recurrence is one."
is intended to mean
"Thus, although it is possible that the billiard table’s initial configuration is one that never recurs, in which case it is a zero probability event; consequently, the billiard table’s initial configuration is one that recurs with probability one and so the probability of recurrence is one."
I'm not trying to change the language, only to add the missing text. The language could improve substantially in this thesis.
Note that the author makes a very basic error in the text that you quote: he seems to think that the opposite of "recurring" is to "never recur". This is a nonsense since points can recur finitely often.