We work in greater generality than in CF1. Let be an ergodic probability measure-preserving system. Let be an abelian locally compact topological group endowed with a metric , its Borel sigma-algebra and its Haar measure. For every , we define the cylinder flow by:
- , and
- defined as .
It is clear that is preserved by . For each , let
be the vertical translation. Because is abelian, commutes with each :
In particular, if is -invariant, then so is .
the above discussion motivates us to introduce the following subgroup of .
Definition 1 Define the collection of periods for as
Saying that is exactly the same as saying that the behavior of is periodic with period . Of course, is a subgroup of , because for any . It is also closed, as we’ll prove below. In particular, if , then is either or for some . In the second situation, we can see as defined in the torus , where for . In the first situation, when , is ergodic. This is indeed the general case, as stated below.
- is a closed subgroup of , and
- is ergodic if and only if .
For the first item, it remains to prove is closed. This follows from the continuous dependence of in : if in , then for any
If in particular each belongs to , then for we have
thus proving that .
We postpone the proof of the second item to Section 2.
Even though the above lemma gives a good criterium for ergodicity, it is not clear when an element belongs to , mainly because the condition has to be checked along elements of only, and this set will eventually contain only the trivial elements of . What we want now is to get rid of this soft description and to define a more quantitative criterium for ergodicity that does not only depend on the elements of . This new notion, which in principle seems to make things more difficult – but does not, is that of
2. Essential values
From now and on, is the ball of of radius centered in , is the -th Birkhoff sum of with respect to and set-theoretical equalities mean set-theoretical equalities modulo zero.
Definition 3 An element is an essential value for if, for any with and any , there is such that
The set of all essential values of is denoted by .
This, on the contrary of the geometrical notion of period, is more quantitative and translates the fact that, in order to be ergodic, the -iterates of every set of positive -measure has to saturate the whole cylinder . As claimed above, this is just an alternative definition for periods, according to the
Proof: Let . By definition, there are with and such that
This implies that if
then for all . This in turn guarantees that if we define by
then . But and so
This proves that .
For the reverse inequality, let . We claim this implies the existence of disjoint with such that
Indeed, take such that and define
For each , define by the equality
Then . Furthermore, the -invariance of gives
and so , thus proving that the map is -invariant. Because is ergodic,
for -almost every .
The idea now is to define a measurable function by
As defined above, may not be measurable, but we can apply a trick – to be explained below – in order to guarantee measurability. Assume we have done this and that as above is measurable. We have and thus, by Lusin’s theorem, there are sets with and such that
We claim the pair contradicts the assumption that . In other words, we claim
which is equivalent to the implication
and this in turn follows if we prove that
And indeed, as , we have
thus establishing (1).
The above proof has two flaws. Firstly, as defined above, may not be measurable. Secondly, we used Lusin’s theorem, which assumes that is defined in a Polish space. We bypass these problems in three steps:
1. In general, may have infinite -measure and then the intersection does not behave measurably. We fix this by considering a measurable subset such that satisfies
2. is not a probability measure, but there exists a probability measure (not necessarily -invariant) equivalent to . Indeed, by the -finiteness of there are disjoint sets with and . The probability measure in defined by
is clearly equivalent to .
3. can be assumed to be Polish. Indeed, if is the measure algebra of (see ERT11),
define a distance in .
We can now define in the probability Polish space by
By Fubini’s theorem, the map is measurable and, by the -finiteness of each (see Exercise 1 below), this new map is measurable. Furthermore, the space of definition of being a Polish space, Lusin’s theorem can be applied to guarantee the existence of the neighborhood in which is bounded away from zero. We leave the details to the reader.
then the maps from to defined by
Lemma 5 (Steinhaus) Let be Borel sets with . Then there is an open set such that
Proof: Obviously, we can assume . Then defined by
is measurable. Note that
By Lusin’s theorem, there is an open set such that for all .
We finally arrive to the proof of the second part of Lemma 2.
Assume is ergodic. For with and , let . For , the ergodicity of guarantees the existence of such that intersects in a set of positive -measure. This implies that
Assume . Let and define by the equality
Using the same argument as in the proof of Theorem 4, for -almost every . For a fixed , we have by assumption that and so for -almost every . Let be a dense subset of . For each , let
and . has zero -measure. We will prove that for every . Assume this is not the case. By Steinhaus’ theorem, there is an open set such that
This is a contradiction: for ,
This completes the proof.
3. An example
In this section we study a particular class of cylinder flow. Let be the Haar function, defined as
Up to my knowledge, Schmidt was the first one to be interested in these cylinder flows, because they encode the properties of deterministic random walks in the following sense: if, instead of the irrational rotation, the basis transformation of is the doubling map, i.e. , then
where is statistically the same as the classical random walk on the integers (starting at zero, at each step we toss a coin and the particle moves one step to the right in case of ‘heads’ and one step to the left in case of `tails’). In this case, the basis transformation has positive topological entropy. In ours, the random walk is driven by a zero topological entropy transformation.
The goal of this section is to prove, via the notion of essential values, that is ergodic. Because is a subgroup of , it is enough to show that . Still with the notation of the previous sections, let , and be the Lebesgue measures in , and , respectively. We first, following the notation of Section 2 of CF1, prove a lemma on continued fractions.
Lemma 6 For every irrational number , there are infinitely many odd positive integers such that
Proof: Let be the continued fraction expansion of and
We use the fact (and leave the proof to the reader) that , where , and consequently
In particular, every odd with satisfies the lemma. By contradiction suppose that, for large , whenever is odd. Fix one such . If is even, is odd and so . Then . If is odd, then both and again .
Proof: We proceed as in the proof of Denjoy-Koksma’s inequality. Consider the partition of in the intervals , . is equal to in of them and to in of them, so the first claim follows if we prove that for each there is one and exactly one integer for which
Let . Because , we can take . Without loss of generality, assume that
Then, for ,
where is the residue of modulo . As runs overs all residues modulo , the first claim is proved. The case is similar, in which case
Now decompose the interval in which changes sign into two subintervals
By the first part, to each there is a unique such that , and if and only if , which happens for an interval of length containing .
Theorem 8 For any irrational number , the cylinder flow is ergodic.
Proof: Let be a sequence of odd positive integers such that . We claim Lemma 7 implies that, for any ,
To each , at least of the intervals are contained in and so
Summing up this in ,
Taking limits in both sides, (2) follows.
Now it is to see that is an essential value. Let with . Because , (2) implies that