Question about $p$-limits on $\beta \omega$ Let $p$ be a free ultrafilter on $\omega$, $X$ be a topological space and $s_n\,(n \in \omega)$ be a sequence. We say that $x \in X$ is a $p$-limit of a $s_n$ if, and only if for every neighborhood $V$ of $x$, $\{n \in \omega: s_n \in V\}\in p$.
It's true that if $X$ is hausdorff, the $p$-limits are unique and that if $X$ is compact, they always exist.
Question: Suppose that $X=\beta \omega=\beta \mathbb N$. Let $s_n, t_n\, (n \in \omega)$ be two sequences on $X$. Is it true that if $\{n \in \omega: s_n\neq t_n\} \in p$ then the $p$-limits of $s_n, t_n$ are distinct?
I know that this is not true for $X$ in general, but I'm interested in the case $X=\beta \mathbb N$.
 A: No, it's possible for two entirely different sequences in $\beta\omega$ to have the same limit with respect to some ultrafilter $p$. A rather silly example is obtained by taking any sequence $(s_n)$ of distinct points in $\beta\omega$ and any non-principal $p\in\beta\omega$, and then letting $(t_n)$ be the constant sequence with all $t_n$ equal to the $p$-limit of $(s_n)$.
But there are also serious examples. Among the most important of these are those arising from idempotent ultrafilters.  A nonoprincipal ultrafilter $p$ on $\omega$ is called idempotent if the sequence of translates $(p+n)_{n\in\omega}$ has $p$-limit equal to $p$ itself.  (Here $p+n$ is defined as the ultrafilter generated by the $n$-translates $A+n=\{a+n:a\in  A\}$ of all the sets $A\in p$.)  Note that any $p\in\beta\omega$ is the $p$-limit of the sequence $(n)_{n\in\omega}$ of principal ultrafilters.  So an idempotent ultrafilter $p$ is simultaneously the $p$-limit of a sequence of principal ultrafilters and of a sequence of nonprincipal ultrafilters.
The existence of idempotent ultrafilters is a nontrivial result (a special case of a theorem of Ellis about existence of idempotent elements in compact semi-topological semigroups). The proof can be found in many places, for example in a survey paper of mine available at http://www.math.lsa.umich.edu/~ablass/ufdyn.pdf (see Theorem 2 and its corollary).  For a great deal more information about idempotent and related ultrafilters and their applications, see the book "Algebra in the Stone-Cech Compactification" by Neil Hindman and Dona Strauss.
