Poincaré Recurrence and Dense Sets This is kind of a spin-off of the question asked here. Take the interval $X:=[0,1]$ with $\mu$ being standard Lebesgue measure. Let $f$ be a measure preserving map $f:[0,1]\rightarrow [0,1]$. The Poincaré Recurrence theorem tells us that if I pick a measurable set $E\subset [0,1]$, then under iterations of $f$ almost every point in $E$ returns to $E$ infinitely often, i.e.
$\mu\left({x\in E: \ \exists N, \mbox{ such that } \forall n\geq N, \ f^n(x)\notin E}\right)=0$
Call the set of exceptions above $M$. My question is:

If I specify an $f$, for what class of sets $E$ in $X$ is $M$ not dense? I am interested in two cases: 

1) "dense" with respect to $X$, if $E$ is dense in $X$
2) "dense" with respect to $E$

For example, let $E=\mathbb{Q}\cap[0,1]$ and $f(E):=E+\phi$ where $\phi$ is irrational. Then $M=E$, which of course is not a contradiction since $\mu(M)=\mu(E)=0$. On the other hand, throw in the set $H:=\{n\cdot \phi: n\in \mathbb{N}\}$, so that $E':=H\cup E$. In this case we still have $M=E$. As well, it looks like the class of sets I'm interested in is $H\cup {\mbox{not a dense set in X}}$.
Note: My original motivation for asking this was to try and conceptualize the Poincaré recurrence theorem for a human physicist. If I were looking at the phase plot of balls on a billiard table at a specific time, I would only be able to give imprecise measurements of both position and velocity. In this case, it seems that in order to invoke Poincaré recurrence, I would need small intervals around every point to recur perfectly, in the sense that $M=\emptyset$. Perhaps this IS the case if $f$ arises from some nice ODE, but I'm interested in a more general setting. I also don't really want to require that $\mu(M)>0$, which is why I feel asking about denseness is more appropriate. 
 A: If $f$ is minimal, i.e. every orbit is dense, then $E$ is either empty or dense. So it remains to decide if your set is empty. Clearly it is non-empty for positive measure sets. If $\mu(E) = 0$. Then also
$$
 \mu(\bigcup_{n} f^{-n} E) =0
$$
and for this set the exceptional set is non-empty (and trivially dense).
A: Theorem
If  $T:X\to X$ transformation which preserves  the measure $\mu$ of a  a probability measure on $(X,\mathcal{B}(X))$. the following statements ara equivalent:
(i) $T$ is ergodic 
(ii) The only members $B$ of $\mathcal{B}$ with $\mu(T^{-1}B\Delta B)=0$ are  those with $\mu(B)=0~~or~~1$
(iii) For every $A\in\mathcal{B}$ with $\mu(A)>0$ we have $\mu(\bigcup_{n\geq 1}T^{n}A)=1$
These characterizations of ergodicity has a clear geometric meaning. But what I want to show you is the following theorem
Theorem
Let be $X$ a compact metric space with enumerable basis , $\mathcal{B}(X)$ the $\sigma$-algebra  of Borel subsets of $X$ ande let $\mu$ be  a probability measure on $(X,\mathcal{B}(X))$ such that $\mu(U)>0$  for all open set $U$ in $X$. Suppose  $T:X\to X$  is  a continuous  transformation which preserves  the measure $¨\mu$  and is ergodic.  Then almost all points of $X$  have a dense  orbit under T,  i.e  $$x\in X: T^{n}(x)  ~~ is~~ dense ~~subset ~~ of ~~X$$ has $\mu$-measure one.
I'm not sure but I think you can do some connection between this theorem and the question you proposed about Poincare Recurrence. Hope this is of some value to you, for more details and proofs of the theorems above search for the book Peter Walters: an introduction to Ergodic Theory.
Ohh man I just remembered this theorem has a topological version!
Let us remember before the definition of   $\omega$-limit. Let be $X$ a topological space and $T:X\to X$ a trasformation. The  $\omega$-limit set of $x\in X$ is defined by (denoted by $\omega(x)$)
$y\in X $ such that for all neighborhood U of $y$  the relation $T^{n}(x)\in U$  is satisfied
for infinite  values of n.
Theorem
Let be $X$ a metric space  and $T:X\to X$ a mensurable transformation  which preserves  the measure $\mu$ of a  a probability measure on $(X,\mathcal{B}(X)).$  Then the set 
$$x: x\notin \omega(x) $$
has 0 $\mu$-measure
This theorem  you will find in the book  the great mathematician Ricardo Mane: Ergodic theory
I'm not sure but I think you can do some connection between this theorems and the question you proposed about Poincare Recurrence. Hope this is of some value to you
