So, now it’s time to connect the topological notions of dynamical systems with ergodic theory (which makes use of measures). Our first example will use the notion of topological transitivity, which we now introduce. The next example will return to the story about recurrent points, which I talked a bit about yesterday.
Say that a homeomorphism of a compact metric space
is topologically transitive if there exists
with
dense in
. (For instance, a minimal homeomorphism is obviously topologically transitive.) Let
be a countable basis for the topology of
. Then the set of all such
(with
dense) is given by
In particular, if it is nonempty, then each is dense—being
-invariant and containing
—and this set is a dense
by Baire’s theorem.
Proposition 1 Let
have a Borel probability measure
positive on every nonempty open set, and let
be measure-preserving and ergodic. Then the set of
with
is of measure 1, so
is topologically transitive.
Indeed, each must have measure zero or one by ergodicity, so measure 1 by hypothesis. Then take the countable intersection.
Poincaré recurrence
We now move to the abstract measure-theoretic framework, not topological.
Theorem 2 (Poincaré recurrence) Let
be a measure-preserving transformation on a probability space
. If
is measurable, then there exists
with
such that for each
, there is a sequence
with
.
In other words, points of are
-frequently in
.
To see this, consider the sets
Then and
since
. Set
and
. Then
and this latter set is the limit of the increasing sequence of sets , each of which has measure zero. This proves the theorem.
Topological application of Poincaré recurrence
Let be a compact metric space,
a continuous map. Say that
is
-recurrent if there exists a sequence
with
. Birkhoff’s theorem states that there always exists a
-recurrent point if
is a homeomorphism.
Theorem 3 If
preserves the probability measure
, then almost all points of
are
-recurrent.
Let be a countable basis. Then
is not
-recurrent if
for some
and
, as is easily seen (check the two cases
a periodic point and
non-periodic separately). So
However, each set in this union has -measure zero by Poincaré’s theorem, which proves this one.
We will show that any continuous has a probability measure
with respect to which
is measure-preserving, so
Corollary 4 If
is a continuous transformation of the compact metric space
, then
-recurrent points exist.
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