— 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 .