in Section 1 is
just the ultrafilter iteration of the shift map, i.e., of the
universal dynamical system. Indeed, we have, for any
ultrafilters 
and 
on 
The addition operation defined for in Section 1 is
just the ultrafilter iteration of the shift map, i.e., of the
universal dynamical system. Indeed, we have, for any
ultrafilters and on
where at the last step we used that 
commutes
with continuous maps, like addition as a function of its
left argument, and that 
.
Of course, this equivalence between iteration in 
and addition allows us to reformulate Theorem 1
algebraically. We do so in the following theorem, adding
some more reformulations in terms of subsemigroups
(i.e., non-empty subsets closed under addition) and ideals
in the semigroup 
. A left ideal is a
non-empty set 
such that if 
and 
then 
; right ideals
and two-sided ideals are defined analogously.
Proof:
Parts (1), (3), and the first part
of (2) are immediate consequences of the
corresponding parts of Theorem 1 and the fact that
.
To finish the proof of (2), notice first that
every ultrafilter generates a left ideal, namely
. It follows that a
minimal left ideal, being the ideal generated by any of its
elements , is closed, for it is the image of the
compact space under the continuous map adding
on the right. That is why ``closed'' is parenthesized
in (2); putting it in or leaving it out doesn't
affect the statement. Now to say that a particular
ultrafilter is in a minimal left ideal is to say that
the left ideal 
that it generates is minimal or,
equivalently, is generated by each of its elements. That
is, for every ultrafilter , the ideal 
generated by 
must be all of .
Equivalently, must contain the generator
of . But that means that, for every
, we can express as 
by
suitably choosing 
. This completes the proof of
(2).
(4), which is included in the theorem
because of its analogy to (2), is trivial in one
direction, as 
is a closed subsemigroup if
is idempotent. To prove the non-trivial direction (due, as
far as I know, to Ellis [14]), let C be a minimal
closed subsemigroup of and let 
. Then
is also closed (being the image of the compact set
C under a continuous map) and a subsemigroup of C,
so by minimality it equals C. In particular it contains
. So the set 
is
nonempty. It is closed (being the pre-image of
under a continuous map) and also a subsemigroup of C,
so it equals C and therefore contains . That is,
, and the proof is complete. (It follows, of
course, by minimality, that 
.)
The information in Theorem 2 about ideals and subsemigroups can be used to give quick existence proofs for the corresponding sorts of ultrafilters.
Proof:
The intersection of a chain of closed subsemigroups of
is again a closed subsemigroup; it is non-empty by
compactness, and it is obviously closed topologically and
closed under addition. By Zorn's Lemma, there are
minimal closed subsemigroups of 
. Their
elements are idempotent by (4) of the
theorem. The same argument applied to closed left ideals
yields uniformly recurrent points.
The concepts characterized in Theorem 2 are related to each other as follows.
Proof:
(1) 
(2)
Let be uniformly recurrent and proximal to 0. By
Theorem 2(3), fix with
. By Theorem 2(2),
fix with
Combining these two equations, we get 
,
and substituting this into the displayed equation we get
.
(2) (3)
If is idempotent, then the requirement
for recurrence and the requirement
for proximality to 0 (see Theorem 2) are
satisfied by taking 
.
The preceding results connect the algebraic properties of
with its dynamical properties, but in fact, thanks to
the universality of among dynamical systems, we
easily get connections between the algebra of and
arbitrary dynamical systems.
Proof:
Each part is proved by combining the corresponding parts
of Theorems 1 and 2 with the fact that 
.
Suppose is recurrent in . So by
Theorem 2(1) there is with
. Then
, so 
is recurrent by Theorem 1(1).
Suppose is uniformly recurrent. By
Theorem 2(2), for every there is
with 
and therefore
. By
Theorem 1(2), is uniformly
recurrent.
Finally, suppose 
and 
are proximal. By
Theorem 2(3), there is with
. Then
. By
Theorem 1(3), 
and
are proximal.
As an application of these connections between dynamics and algebra, we give a short proof of the Auslander-Ellis Theorem [15].
Proof:
By the corollary of Theorem 2, there exists a uniformly
recurrent . It follows immediately that
every ultrafilter of the form 
is uniformly
recurrent. The set 
of such ultrafilters is a
closed subsemigroup of . By Zorn's Lemma, it
includes a minimal closed subsemigroup. By
Theorem 2(4), there is an idempotent
. Then , being uniformly
recurrent and idempotent, is also proximal to 0 by
Theorem 3.
Now for X, T, and x as in the theorem, let
. Then, by Theorem 4, y is uniformly
recurrent and proximal to 
.
The property of ultrafilters, ``uniformly recurrent and proximal to 0,'' which played a key role in the proof of Theorem 5, has alternative algebraic descriptions that will be useful later. To introduce them, we first define a partial ordering of the idempotent ultrafilters by
This definition and parts of the next theorem are from [4]. When we refer to an idempotent ultrafilter as minimal, we mean with respect to this ordering.
Proof:
The equivalence of (1) and (2) is immediate from Theorem 2(2) and Theorem 3.
To prove (2) (3),
assume (2), and suppose is an
idempotent 
. Since is uniformly
recurrent by Theorem 2, choose so that
, which reduces, in view of
, to 
. Using this, the
idempotence of , and again , we
compute
so is minimal.
We next show that, if is idempotent and
is a minimal left ideal, then there is
an idempotent in I. Since we already
know that (2) implies (3), this
gives the last sentence of the theorem; it will also be
useful in establishing
(3) (2). So let such
and I be given. Being a closed subsemigroup of
, I contains an idempotent by the same argument as in the
proof of Theorem 5. Being in
, this satisfies 
because is idempotent. Let 
. Then
belongs to the left ideal I because does.
From and the idempotence of
and , we infer
and
which mean that , as desired.
The proof of (3) (2) is
now easy. If is a minimal idempotent, apply Zorn's
Lemma to get a minimal left ideal
as in the preceding paragraph, and let be obtained
as there. Being , this must be equal to
by minimality. So .
Finally, we must prove that every ultrafilter satisfying
(2) belongs to every two-sided ideal. In
fact, every minimal left ideal I is included in every
two-sided ideal J. To see this, let and
. Then 
is non-empty because it
contains . So is a left ideal, and it
must equal I because I is minimal. So 
.