The horocycle flow is mixing of all orders

— 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 {{\mathbb R}}-actions is given below in Definition 2; we begin by describing the horocycle flow. Let {SL(2,{\mathbb R})} denote the set of all {2\times2} matrices with real entries and determinant {1}, endowed with the usual topology and the (left and right invariant) Haar measure {\lambda}. Given a discrete subgroup {\Gamma\subset SL(2,{\mathbb R})}, the quotient {X:=SL(2,{\mathbb R})/\Gamma} is given the quotient topology and `quotient Haar measure’ {\mu}. The latter can be described by

\displaystyle \int_X\sum_{\gamma\in\Gamma}f(x\gamma)~\mathtt{d}\mu(x)=\int_{SL(2,{\mathbb R})}f(g)~\mathtt{d}\lambda(g)

for any {f\in C_c(SL(2,{\mathbb R}))}. If the total measure {\mu(X)} of {X} is finite (in which case we normalize it so that {\mu(X)=1}), we say that {\Gamma} is a lattice. The classical example is when {\Gamma=SL(2,{\mathbb Z})}.

The horocycle flow is the (continuous) {{\mathbb R}}-action on {X} defined by {s\cdot(g\Gamma)=(h_sg)\Gamma}, where {h_s=\left[\begin{array}{cc} 1 & s \\ 0 & 1 \end{array}\right]} for {s\in{\mathbb R}} and {g\in SL(2,{\mathbb R})}. We will also need the geodesic flow, corresponding in this case to {g_t=\left[\begin{array}{cc} e^{t/2} & 0 \\ 0 &e^{-t/2} \end{array}\right]}.

The theorem we will prove actually deals with a more general situation:

Theorem 1 Let {X} be a (finite dimensional) manifold, let {\mu} a Borel probability measure on {X} and let {(h_s)_{s\in{\mathbb R}}} and {(g_t)_{t\in{\mathbb R}}} be continuous {\mu}-preserving {{\mathbb R}}-actions. If {(h_s)_{s\in{\mathbb R}}} is ergodic and there exists {\lambda>1} such that

\displaystyle  g_th_sg_t^{-1}=h_{s\lambda^t}, \ \ \ \ \ (1)

then {(h_s)_{s\in{\mathbb R}}} is mixing of all orders.

The first step of the proof of Theorem 1 is to show that {(h_s)_{s\in{\mathbb R}}} satisfies a weaker property, called property {P(N)} (see Definition 3) which resembles a property equivalent to weakly mixing of order {N} (at least in the case of {{\mathbb Z}}-actions). For this step, all we need is that {(h_s)_{s\in{\mathbb R}}} is a continuous probability preserving mixing {{\mathbb R}}-action .

— 2. Notions of mixing —

In this section I motivate, state and compare some notions of mixing for probability preserving {{\mathbb R}}-actions. A secondary purpose of this section is to familiarize the reader with the notation for {{\mathbb R}}-actions.

The most basic notion of mixing is simply ergodicity. According to the mean ergodic theorem, a probability preserving system {(X,{\mathcal B},\mu,(h_s)_{s\in{\mathbb R}})} satisfies this property if and only if

\displaystyle \forall f\in L^2(X)\qquad\lim_{N-M\rightarrow\infty}\frac1{N-M}\int_M^Nh_sf~\mathtt{d}s=\int_Xf~\mathtt{d}\mu\qquad\text{in }L^2 \ \ \ \ \ (2)

Here and in the rest of this post we use (as a slight abuse of language) the notation {h_sf} to denote the function {f\circ h_s}. The convergence in (2) is in the strong topology of the Hilbert space {L^2(X)}, which trivially implies convergence in the weak topology. Not so immediate is the fact that the converse is also true, i.e. the system {(X,{\mathcal B},\mu,(h_s)_{s\in{\mathbb R}})} is ergodic if and only if

\displaystyle \forall f,g\in L^2(X)\quad\lim_{N-M\rightarrow\infty}\frac1{N-M}\int_M^N\int_Xh_sf\cdot g~\mathtt{d}\mu~\mathtt{d}s=\int_Xf~\mathtt{d}\mu\int_Xg~\mathtt{d}\mu \ \ \ \ \ (3)

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:

\displaystyle \forall f,g\in L^2(X)\quad\lim_{N-M\rightarrow\infty}\frac1{N-M}\int_M^N\left|\int_Xh_sf\cdot g\mathtt{d}\mu\mathtt{d}s-\int_Xf\mathtt{d}\mu\int_Xg\mathtt{d}\mu\right|=0

Using regular limits, we obtain the notion of strong mixing (or just mixing):

\displaystyle \forall f,g\in L^2(X)\qquad\lim_{s\rightarrow\infty}\int_Xh_sf\cdot g~\mathtt{d}\mu=\int_Xf~\mathtt{d}\mu\int_Xg~\mathtt{d}\mu \ \ \ \ \ (4)

In words, equation (4) states that for any function {f\in L^2}, the orbit {h_sf} converges to the constant function {\int fd\mu} 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 {\|h_sf\|_{L^2}=\|f\|_{L^2}} for all {s\in{\mathbb R}}, and hence any non-zero function {f\in L^2(X)} with {\int fd\mu=0} 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 {\int_Xh_sf_1\cdot h_uf_2\cdot f_3~\mathtt{d}\mu}, when {s,u,s-u} all go to infinity. In general we have the notion of strong mixing of order {N}.

Definition 2 (Strong mixing of order {N}) Let {(X,{\mathcal B},\mu,(h_s)_{s\in{\mathbb R}})} be a probability preserving system and let {N\in{\mathbb N}}. We say that the system is strongly mixing of order {N} if for any {f_0,\dots,f_N\in L^\infty(X)} and any sequences {s_0,\dots,s_N:{\mathbb N}\rightarrow{\mathbb R}} satisfying {s_i(\ell)-s_{i-1}(\ell)\rightarrow\infty} as {\ell\rightarrow\infty} for any {i=1,\dots,N} we have

\displaystyle \lim_{\ell\rightarrow\infty}\int_Xh_{s_0(\ell)}f_0\cdots h_{s_N(\ell)}f_N~\mathtt{d}\mu=\int_Xf_0~\mathtt{d}\mu\cdots\int_Xf_N~\mathtt{d}\mu\ \ \ \ \ (5)

We say that the system is strongly mixing of all orders if it is strong mixing of order {N} for every {N\in{\mathbb N}}.

For each {N\in{\mathbb N}}, it is currently unknown whether strong mixing of order {N} implies strong mixing of order {N+1}. Observe that when {N=1}, (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

\displaystyle \lim_{N-M\rightarrow\infty}\frac1{N-M}\int_M^Nh_{s_1(u)}f_1\cdots h_{s_N(u)}f_N~du=\int_Xf_1~d\mu\cdots\int_Xf_N~d\mu\quad\text{in }L^2

One issue with this definition is that the average is over {u}, but the orbits are at times {s_i(u)}. Thus one is comparing an interval of size {s_1(N)-s_1(M)} in the orbit of {f_1} with an interval of size {s_2(N)-s_2(M)} in the orbit of {f_2}.

To motivate the next definition, we first look at an equivalent way to express strong mixing of order {N}. If we let {K_i=s_i/s_N} it’s not hard to see that a system is strongly mixing of order {N} if and only if for any {f_0,\dots,f_N\in L^\infty(X)}, any sequences {K_0,K_1,\dots,K_N:{\mathbb N}\rightarrow{\mathbb R}} satisfying {0\equiv K_0\leq K_1\leq\cdots<K_N\equiv1} and {\big(K_i(\ell)-K_{i-1}(\ell)\big)\ell\rightarrow\infty} as {\ell\rightarrow\infty} for {i=1,\dots,N} we have

\displaystyle \lim_{\ell\rightarrow\infty}\int_Xf_0h_{K_1(\ell)\ell}f_1\cdots h_{K_{N-1}(\ell)\ell}f_{N-1} h_\ell f_Nd\mu=\int_Xf_0d\mu\cdots\int_Xf_Nd\mu\ \ \ \ \ (6)

One can now formulate an analogue of (6) in the strong topology, but using a Cesàro limit.

Definition 3 (Property {P(N)}) Let {(X,{\mathcal B},\mu,(h_s)_{s\in{\mathbb R}})} be a probability preserving system and let {N\in{\mathbb N}}. We say that the system satisfies the property {P(N)} if for any {f_1,\dots,f_N\in C_c(X)} and any sequences {m,n,K_0,\dots,K_N:{\mathbb N}\rightarrow{\mathbb R}} satisfying

  • {n(\ell)-m(\ell)\rightarrow\infty} as {\ell\rightarrow\infty},
  • For all {\ell\in{\mathbb N}}, {0=K_0(\ell)<K_1(\ell)<\cdots<K_N(\ell)=1},
  • For {i=1,\dots,N} we have {\big(K_i(\ell)-K_{i-1}(\ell)\big)n(\ell)\rightarrow\infty} as {\ell\rightarrow\infty},

we have

\displaystyle  \lim_{\ell\rightarrow\infty}\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\prod_{i=1}^Nh_{K_i(\ell)u}f_i~\mathtt{d}u= \prod_{i=1}^N\int_Xf_i~\mathtt{d}\mu\qquad\text{in }L^2 \ \ \ \ \ (7)

Observe that when {N=1}, (7) reduces to (2). Hence {P(1)} is equivalent to ergodicity.

— 3. Mixing implies {P(N)} for all {N\in{\mathbb N}}. —

In this section we prove that the horocycle flow (and indeed any mixing continuous probability preserving {{\mathbb R}}-action) satisfies the property {P(N)} for every {N\in{\mathbb N}}. The proof proceeds by induction on {N} . 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.

Lemma 4 [Robust van der Corput trick] Let {H} be a Hilbert space, let {(\phi_{\ell,u})_{\ell\in{\mathbb N},u\in{\mathbb R}}\subset H} be a bounded family of vectors in {H}, depending continuously on {u}, and let {n,m:{\mathbb N}\rightarrow{\mathbb R}} be sequences such that {n(\ell)-m(\ell)\rightarrow\infty} as {\ell\rightarrow\infty}. If

\displaystyle \lim_{D\rightarrow\infty}\frac1D\int_0^D\limsup_{\ell\rightarrow\infty}\left|\frac1{n(\ell)-m(\ell)} \int_{m(\ell)}^{n(\ell)}\langle\phi_{\ell,u},\phi_{\ell,u+d}\rangle~\mathtt{d}u\right|\mathtt{d}d=0,\ \ \ \ \ (8)


\displaystyle \lim_{\ell\rightarrow\infty}\left\|\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\phi_{\ell,u}~\mathtt{d}u\right\|=0

where the integral is in the Bochner sense.

Proof: Without loss of generality, we will assume that {\|\phi_{\ell,u}\|\leq1} for all {\ell,u}. Then for each {d>0} and {n>m} we have

\displaystyle \left\|\int_m^n\phi_{\ell,u}~\mathtt{d}u- \int_m^n\phi_{\ell,u+d}~\mathtt{d}u\right\|<2d

and hence for {D>0}

\displaystyle \left\|\frac1{n-m}\int_m^n\phi_{\ell,u}~\mathtt{d}u- \frac1D\int_0^D\frac1{n-m}\int_m^n\phi_{\ell,u+d}~\mathtt{d}u~\mathtt{d}d\right\|<\frac{2D}{n-m} \ \ \ \ \ (9)

Fix {\epsilon>0} arbitrary and take a large {D} and {\ell}, depending on {\epsilon}. More precisely, take {D_0} so that for any {D>D_0}

\displaystyle \frac1D\int_0^D\limsup_{\ell\rightarrow\infty}\left|\frac1{n(\ell)-m(\ell)} \int_{m(\ell)}^{n(\ell)}\langle\phi_{\ell,u},\phi_{\ell,u+d}\rangle~\mathtt{d}u\right|\mathtt{d}d<\epsilon\ \ \ \ \ (10)

Next let {D=D_0/\epsilon} and define the sequence of functions {(f_\ell)_{\ell\in{\mathbb N}}} on {[D_0,D]} by

\displaystyle f_\ell(D')=\frac1{D'}\int_0^{D'}\left|\frac1{n(\ell)-m(\ell)} \int_{m(\ell)}^{n(\ell)}\langle\phi_{\ell,u},\phi_{\ell,u+d}\rangle~\mathtt{d}u\right|~\mathtt{d}d

It follows from Fatou’s lemma and (10) that {\limsup_{\ell\rightarrow\infty}f_\ell(D')<\epsilon} for each {D'\in[D_0,D]}. Moreover, it is not hard to check that {|f_\ell(D_1)-f_\ell(D_2)|\leq|D_1-D_2|}, so we can choose {\ell} so that {n(\ell)-m(\ell)>D/\epsilon} and

\displaystyle \forall D'\in[D_0,D]\qquad\frac1{D'}\int_0^{D'}\left|\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\langle \phi_{\ell,u},\phi_{\ell,u+d}\rangle~\mathtt{d}u\right|~\mathtt{d}d<\epsilon\ \ \ \ \ (11)

In view of (9), it suffices to show that

\displaystyle \left\|\frac1D\int_0^D\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\phi_{\ell,u+d}~\mathtt{d}u~\mathtt{d}d\right\| <8\epsilon

From now on until the end of the proof {\ell} is fixed, so we will omit it (i.e., we will write {m,n,\phi_u} instead of {m(\ell),n(\ell),\phi_{\ell,u}}). Finally, using Jensen’s inequality we compute

\begin{array}{rl}  &\displaystyle \left\|\frac1D\int_0^D\frac1{n-m}\int_m^n \phi_{u+d}~\mathtt{d}u~\mathtt{d}d\right\|^2 \\ \text{(Jensen's inequality)}\leq~&~\displaystyle \frac1{n-m}\int_m^n\left\|\frac1D\int_0^D \phi_{u+d}~\mathtt{d}d\right\|^2~\mathtt{d}u \\=~&~\displaystyle \frac1{n-m}\int_m^n\frac1D\int_0^D\frac1D\int_0^D \langle\phi_{u+d_1},\phi_{u+d_2}\rangle~\mathtt{d}d_2~\mathtt{d}d_1~\mathtt{d}u \\=~&~\displaystyle \frac1{n-m}\int_m^n\frac1D\int_0^D\frac1D\int_{d_1}^D 2\,\mathrm{Re}\langle\phi_{u+d_1},\phi_{u+d_2}\rangle~\mathtt{d}d_2~\mathtt{d}d_1~\mathtt{d}u \\=~&~\displaystyle 2\,\mathrm{Re}\frac1D\int_0^D\frac1D\int_{d_1}^D\frac1{n-m}\int_{m+d_1}^{n+d_1} \langle\phi_u,\phi_{u+d_2-d_1}\rangle~\mathtt{d}u~\mathtt{d}d_2~\mathtt{d}d_1 \\=~&~\displaystyle 2\,\mathrm{Re}\frac1D\int_0^D\frac1D\int_0^{D-d_1}\frac1{n-m}\int_{m+d_1}^{n+d_1} \langle\phi_u,\phi_{u+d_3}\rangle~\mathtt{d}u~\mathtt{d}d_3~\mathtt{d}d_1 \\ \leq~&~\displaystyle \frac{4D}{n-m}+2\,\mathrm{Re}\frac1D\int_0^D\frac1D\int_0^{D-d_1}\left|\frac1{n-m}\int_m^n \langle\phi_u,\phi_{u+d_3}\rangle~\mathtt{d}u\right|~\mathtt{d}d_3~\mathtt{d}d_1 \\ \leq~&~\displaystyle 4\epsilon+\frac{2D_0}D+2\,\mathrm{Re}\frac1D\int_0^{D-D_0}\frac1D\int_0^{D-d_1}\left|\frac1{n-m}\int_m^n \langle\phi_u,\phi_{u+d_3}\rangle~\mathtt{d}u\right|~\mathtt{d}d_3~\mathtt{d}d_1 \\ \leq~&~\displaystyle 4\epsilon+2\epsilon+2\epsilon~\leq~8\epsilon \end{array}

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 {f_i} to also depend on {\ell}, as long as they ultimately converge.

Definition 5 (Robust property {P}) Let {(X,{\mathcal B},\mu,(h_s)_{s\in{\mathbb R}})} be a probability preserving system and let {N\in{\mathbb N}}. We say that the system satisfies the property {RP(N)} if for any {f_{1,\ell},\dots,f_{N,\ell}} and any sequences {m,n,K_0,\dots,K_N:{\mathbb N}\rightarrow[0,1]} satisfying

  • For each {i=1,\dots,N} and {\ell\in{\mathbb N}} we have {f_{i,\ell}\in C_c(X)}.
  • For each {i=1,\dots,N}, there exists some {f_i\in C_c(X)} such that {f_{i,\ell}\rightarrow f_i} as {\ell\rightarrow\infty} in {C_c(X)}.
  • {n(\ell)-m(\ell)\rightarrow\infty} as {\ell\rightarrow\infty},
  • For all {\ell\in{\mathbb N}}, {0=K_0(\ell)<K_1(\ell)<\cdots<K_N(\ell)=1},
  • For {i=1,\dots,N} we have {\big(K_i(\ell)-K_{i-1}(\ell)\big)n(\ell)\rightarrow\infty} as {\ell\rightarrow\infty},

we have

\displaystyle  \lim_{\ell\rightarrow\infty}\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\prod_{i=1}^Nh_{K_i(\ell)u}f_{i,\ell}~\mathtt{d}u =\prod_{i=1}^N\int_Xf_i~\mathtt{d}\mu\qquad\text{in }L^2 \ \ \ \ \ (12)

Since Proposition 6 holds for a general continuous mixing flow, I use {T} to denote the flow. In this post we only need this proposition for {T=h}.

Proposition 6 If a flow is mixing (i.e., satisfies (4)), then {RP(N-1)\Rightarrow RP(N)}. Therefore a mixing system satisfies {RP(N)} for every {N}.

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 {RP(N-1)} holds. We first pass to a subsequence so that the limit {\tilde K_i=\lim K_i(\ell)} exists for each {i}. When {f_N} is a constant, it is not hard to see that (12) follows directly from {RP(N-1)}. Therefore we can assume that {\int_Xf_Nd\mu=0} (and hence the right hand side of (12) is {0}). In this case, letting

\displaystyle \phi_{\ell,u}=\prod_{i=1}^NT_{K_i(\ell)u}f_{i,\ell},

we see that our goal (12) becomes the conclusion of Lemma 4. Therefore we only need to show that (8) holds. Fixing {d,u} and {\ell}, we compute

\displaystyle  \begin{array}{rcl} \displaystyle \langle \phi_{\ell,u+d},\phi_{\ell,u}\rangle&=&\displaystyle \int_X\prod_{i=1}^NT_{K_i(\ell)(u+d)}f_{i,\ell}\prod_{i=1}^NT_{K_i(\ell)u}f_{i,\ell}~\mathtt{d}\mu\\&=&\displaystyle \int_X\prod_{i=1}^NT_{K_i(\ell)u}\big(f_{i,\ell}\cdot T_{K_i(\ell)d}f_{i,\ell}\big)~\mathtt{d}\mu\\&=&\displaystyle \int_Xf_{1,\ell}\cdot T_df_{1,\ell}\cdot\prod_{i=2}^NT_{\big(K_i(\ell)-K_1(\ell)\big)u}\big(f_{i,\ell}\cdot T_{K_i(\ell)d}f_{i,\ell}\big)~\mathtt{d}\mu\\ \end{array}

Now let {\psi_{i,\ell}=f_{i,\ell}\cdot T_{K_i(\ell)d}f_{i,\ell}}. Since, for each {i=1,\dots,N}, the sequence {K_i(\ell)\rightarrow\tilde K_i} and {f_{i,\ell}\rightarrow f_i} as {\ell\rightarrow\infty} in {C_c(X)}, we deduce that {\psi_{i,\ell}\rightarrow\psi_i:=f_i\cdot T_{\tilde K_id}f_i} as {\ell\rightarrow\infty} in {C_c(X)} as well. Thus, using {RP(N-1)} we deduce that

\displaystyle  \lim_{\ell\rightarrow\infty}\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\langle \phi_{\ell,u+d},\phi_{\ell,u}\rangle\mathtt{d}u=\prod_{i=1}^N\int_Xf_i\cdot T_{\tilde K_id}f_i~\mathtt{d}\mu \ \ \ \ \ (13)

Recall that {\int_Xf_N~\mathtt{d}\mu=0} and {\tilde K_N=1}. Thus it follows from (4) that the {N}-th term in the product converges to {0} as {d\rightarrow\infty}. All the other terms in the product are bounded; therefore the (Cesàro) limit of the product in the right hand side is also {0}. This shows that (8) holds and this finishes the proof. \Box

— 4. {P(N)~~+} (1) imply mixing of order {N}

In this section we prove the following result.

Proposition 7 Under the assumptions of Theorem 1, if {(h_s)_{s\in{\mathbb R}}} satisfies {P(N)}, then it is mixing of order {N}.

Before we prove Proposition 7, we see how, together with Proposition 6, it implies Theorem 1.

Proof: } Observe that {P(1)} is equivalent to ergodicity, hence {(h_s)_{s\in{\mathbb R}}} satisfies {P(1)}. Now Proposition 7 implies that {(h_s)_{s\in{\mathbb R}}} is strong mixing. It follows from Proposition 6 that {(h_s)_{s\in{\mathbb R}}} satisfies {P(N)} for all {N\in{\mathbb N}}. Finally, appealing once more to Proposition 7 we conclude that {(h_s)_{s\in{\mathbb R}}} is mixing of all orders. \Box

All that remains to show now is Proposition 7. We need the following notion.

Definition 8 A local section for the flow {(g_t)_{t\in{\mathbb R}}} is a compact subset {\Sigma\subset X} for which there exists {a>0} such that the map {\phi:[0,a]\times\Sigma\rightarrow X} defined by {\phi(t,y)=g_ty} is a homeomorphism onto its image. The image of {\phi}, denoted by {A(a,\Sigma)}, is called a flowbox.

The following two technical lemmas explain why flowboxes are useful.

Lemma 9 For each local section {\Sigma} there exists a unique Borel measure {\mu_\Sigma} on {\Sigma} such that, for any {a} for which {A(a,\Sigma)} is a flowbox, we have

\displaystyle \phi_*\big(l|_{[0,a]}\times\mu_\Sigma\big)=\mu|_{A(a,\Sigma)}

where {l} is the (non-normalized) Lebesgue measure, {\phi} is the map from Definition 8 and {\phi_*} denotes the push forward of a measure under {\phi}.

The measure {\mu_\Sigma} can be defined by {\mu_\Sigma(B)=\mu\Big(\phi\big([0,a]\times B\big)\Big)/a} for every measurable {B\subset\Sigma}. It is not difficult to show that this does not depend on {a} (as long as it is small enough so that {\phi} 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 {X}, and not an arbitrary space.

Lemma 10 Let {a_0>0}. The family of all flowboxes {A(a,\Sigma)} for {(g_t)_{t\in{\mathbb R}}} with {\mu_\Sigma(\Sigma)\leq1} and {a<a_0} generates the Borel {\sigma}-algebra on {X}.

We can now prove Proposition 7.

Proof: In order to prove (5), and in view of Lemma 10, it suffices to assume that {f_0} is the indicator function of a flowbox {A(a,\Sigma)} with {\mu_\Sigma(\Sigma)\leq1} and {a} arbitrarily small ({a} can depend on {f_1,\dots,f_n}, and {\lambda} from (1), but nothing else, including the {s_i}). We will also assume that each {f_i} is in {C_c(X)} and is bounded in absolute value by {1}. Finally, replacing each {s_i} with {s_i-s_0} if necessary (and using the fact that {(h_s)_{s\in{\mathbb R}}} preserves {\mu}) we can assume that {s_0=0}.

Using Lemma 9, the space average in the left hand side of (5) can be rewritten as

\displaystyle \int_X\prod_{i=0}^Nh_{s_i}f_i~\mathtt{d}\mu= \int_0^a\int_\Sigma\prod_{i=1}^Nf_i\big(h_{s_i}g_ty\big)~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}t

Using the relation between {g_t} and {h_s} given by (1) this expression becomes

\displaystyle \int_0^a\int_\Sigma\prod_{i=1}^Nf_i\big(g_th_{\lambda^{-t}s_i}y\big)~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}t

Since each {f_i\in C_c(X)} and {(g_t)_{t\in{\mathbb R}}} is a continuous action, for small enough {a}, the effect of {g_t} in the above integral will be very small. In other words:

\displaystyle \int_X\prod_{i=0}^Nh_{s_i}f_i~\mathtt{d}\mu=\int_0^a\int_\Sigma\prod_{i=1}^Nf_i\big(h_{\lambda^{-t}s_i}y\big)~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}t+o_{a\rightarrow0}(1)

Next we make the change of variable {u=\lambda^{-t}s_N} and obtain

\displaystyle  \begin{array}{rcl}  &&\displaystyle \int_0^a\int_\Sigma\prod_{i=1}^Nf_i(h_{\lambda^ts_i}y)~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}t\\ &=&\displaystyle \int^{s_N}_{\lambda^{-a}s_N}\frac1{u\log\lambda}\int_\Sigma\prod_{i=1}^Nf_i(h_{us_i/s_N}y) ~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}u\\&=&\displaystyle \frac{1-\lambda^{-a}}{a\log\lambda}\cdot\frac{s_N}{u_0}\cdot\frac a{s_N-\lambda^{-a}s_N} \int^{s_N}_{\lambda^{-a}s_N}\int_\Sigma\prod_{i=1}^Nf_i(h_{us_i/s_N}y) ~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}u \end{array}

for some {u_0\in[\lambda^{-a}s_N,s_N]}. Note that the first two factors in the last expression are very close to {1} when {a} is small (independently on any {s_i}). Thus all we need to show is that

\displaystyle  \frac a{s_N(\ell)-\lambda^{-a}s_N(\ell)} \int^{s_N(\ell)}_{\lambda^{-a}s_N(\ell)}\int_\Sigma\prod_{i=1}^Nf_i(h_{us_i(\ell)/s_N(\ell)}y) ~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}u\xrightarrow[\ell\rightarrow\infty]{ }\prod_{i=0}^N\int_Xf_i\mathtt{d}\mu\quad (14)

To ease the notation, let {n(\ell)=s_N(\ell)}, {m(\ell)=\lambda^{-a}s_N(\ell)} and {K_i(\ell)=s_i(\ell)/s_N(\ell)} for each {i=1,\dots,N}. Finally let

\displaystyle \Phi_\ell(t)=\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\int_\Sigma\prod_{i=1}^Nf_i(h_{K_i(\ell)u}g_ty)~\mathtt{d}\mu_\Sigma(y)~\mathtt{d}u

Then (14) becomes

\displaystyle  \lim_{\ell\rightarrow\infty}\Phi_\ell(0)=\mu_\Sigma(\Sigma)\cdot\prod_{i=1}^N\int_Xf_i~\mathtt{d}\mu \ \ \ \ \ (15)

It is easy to see from (1) (and a change of variable) that {(\Phi_\ell)_{\ell\in{\mathbb N}}} is an equicontinuous family at {t=0}. Therefore (15) follows from

\displaystyle  \forall b>0\qquad\lim_{\ell\rightarrow\infty}\frac1b\int_0^b\Phi_\ell(t)~\mathtt{d}t\rightarrow\mu_\Sigma(\Sigma)\cdot\prod_{i=1}^N\int_Xf_i~\mathtt{d}\mu \ \ \ \ \ (16)

Finally, since

\displaystyle \int_0^b\Phi_\ell(t)~\mathtt{d}t=\int_X1_{A(b,\Sigma)}\frac1{n(\ell)-m(\ell)}\int_{m(\ell)}^{n(\ell)}\prod_{i=1}^Nh_{K_i(\ell)u}f_i~\mathtt{d}u~\mathtt{d}\mu,

the limit in (16) follows directly from the property {P(N)}. This finishes the proof. \Box

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