— 1. Introduction —
The main purpose of this post is to present a proof, due to Brian Marcus, that the horocycle flow is mixing of all orders. The precise definition of mixing of all orders for -actions is given below in Definition 2; we begin by describing the horocycle flow. Let denote the set of all matrices with real entries and determinant , endowed with the usual topology and the (left and right invariant) Haar measure . Given a discrete subgroup , the quotient is given the quotient topology and `quotient Haar measure’ . The latter can be described by
for any . If the total measure of is finite (in which case we normalize it so that ), we say that is a lattice. The classical example is when .
The horocycle flow is the (continuous) -action on defined by , where for and . We will also need the geodesic flow, corresponding in this case to .
The theorem we will prove actually deals with a more general situation:
then is mixing of all orders.
The first step of the proof of Theorem 1 is to show that satisfies a weaker property, called property (see Definition 3) which resembles a property equivalent to weakly mixing of order (at least in the case of -actions). For this step, all we need is that is a continuous probability preserving mixing -action .
— 2. Notions of mixing —
In this section I motivate, state and compare some notions of mixing for probability preserving -actions. A secondary purpose of this section is to familiarize the reader with the notation for -actions.
The most basic notion of mixing is simply ergodicity. According to the mean ergodic theorem, a probability preserving system satisfies this property if and only if
Here and in the rest of this post we use (as a slight abuse of language) the notation to denote the function . The convergence in (2) is in the strong topology of the Hilbert space , which trivially implies convergence in the weak topology. Not so immediate is the fact that the converse is also true, i.e. the system is ergodic if and only if
One can arrive at stronger notions of mixing by replacing the (uniform) Cesàro convergence with stronger modes of convergence. For instance, using strong Cesàro convergence (discussed in a previous post) we obtain the notion of weak mixing:
In words, equation (4) states that for any function , the orbit converges to the constant function in the weak topology. The analogue statement with convergence in the strong topology would be rather trivial (only the one point system would satisfy it!), because for all , and hence any non-zero function with would contradict such a strong form of `mixing’.
To get stronger forms of mixing than (4), one can consider higher order correlations, for instance of the form , when all go to infinity. In general we have the notion of strong mixing of order .
We say that the system is strongly mixing of all orders if it is strong mixing of order for every .
For each , it is currently unknown whether strong mixing of order implies strong mixing of order . Observe that when , (5) reduces to (4). In the same way that (4) is a strengthening of (3), which is then equivalent to (2), we can find a weakening of (5) using a Cesàro limit but the strong topology.
The most straightforward way to do this would arguably be to replace (5) with
One issue with this definition is that the average is over , but the orbits are at times . Thus one is comparing an interval of size in the orbit of with an interval of size in the orbit of .
To motivate the next definition, we first look at an equivalent way to express strong mixing of order . If we let it’s not hard to see that a system is strongly mixing of order if and only if for any , any sequences satisfying and as for we have
One can now formulate an analogue of (6) in the strong topology, but using a Cesàro limit.
- as ,
- For all , ,
- For we have as ,
— 3. Mixing implies for all . —
In this section we prove that the horocycle flow (and indeed any mixing continuous probability preserving -action) satisfies the property for every . The proof proceeds by induction on . The main idea is to adapt a trick of Furstenberg, showing that a weak mixing system is weak mixing of all orders, in a certain sense. I have presented a modern version of this trick in a previous post, using the so-called van der Corput trick.
In this post I will adapt the van der Corput trick to this situation.
where the integral is in the Bochner sense.
Proof: Without loss of generality, we will assume that for all . Then for each and we have
Next let and define the sequence of functions on by
In view of (9), it suffices to show that
From now on until the end of the proof is fixed, so we will omit it (i.e., we will write instead of ). Finally, using Jensen’s inequality we compute
where in the last step we used (11).
In order to prove the Property P using the van der Corput trick, we need to bootstrap by induction a stronger property which we call robust property P. The difference is that we now allow the functions to also depend on , as long as they ultimately converge.
- For each and we have .
- For each , there exists some such that as in .
- as ,
- For all , ,
- For we have as ,
Since Proposition 6 holds for a general continuous mixing flow, I use to denote the flow. In this post we only need this proposition for .
Proposition 6 If a flow is mixing (i.e., satisfies (4)), then . Therefore a mixing system satisfies for every .
In fact it suffices to have a weakening of (4) with a strong Cesàro convergence instead of regular convergence (i.e. weak mixing).
Proof: Assume everything is given as in Definition 5 and that holds. We first pass to a subsequence so that the limit exists for each . When is a constant, it is not hard to see that (12) follows directly from . Therefore we can assume that (and hence the right hand side of (12) is ). In this case, letting
Recall that and . Thus it follows from (4) that the -th term in the product converges to as . All the other terms in the product are bounded; therefore the (Cesàro) limit of the product in the right hand side is also . This shows that (8) holds and this finishes the proof.
— 4. (1) imply mixing of order —
In this section we prove the following result.
Proposition 7 Under the assumptions of Theorem 1, if satisfies , then it is mixing of order .
Proof: } Observe that is equivalent to ergodicity, hence satisfies . Now Proposition 7 implies that is strong mixing. It follows from Proposition 6 that satisfies for all . Finally, appealing once more to Proposition 7 we conclude that is mixing of all orders.
All that remains to show now is Proposition 7. We need the following notion.
The following two technical lemmas explain why flowboxes are useful.
where is the (non-normalized) Lebesgue measure, is the map from Definition 8 and denotes the push forward of a measure under .
The measure can be defined by for every measurable . It is not difficult to show that this does not depend on (as long as it is small enough so that is injective).
Lemma 10 seems to be the only place in the proof where we use the fact that the flow runs on a manifold , and not an arbitrary space.
We can now prove Proposition 7.
Proof: In order to prove (5), and in view of Lemma 10, it suffices to assume that is the indicator function of a flowbox with and arbitrarily small ( can depend on , and from (1), but nothing else, including the ). We will also assume that each is in and is bounded in absolute value by . Finally, replacing each with if necessary (and using the fact that preserves ) we can assume that .
Using the relation between and given by (1) this expression becomes
Since each and is a continuous action, for small enough , the effect of in the above integral will be very small. In other words:
Next we make the change of variable and obtain
To ease the notation, let , and for each . Finally let
Then (14) becomes
the limit in (16) follows directly from the property . This finishes the proof.