Exponential decay of correlations

Let {(X,\mu)} be a probability space and {f\colon X\rightarrow X} a measure-preserving transformation. Let {B_1,B_2} be Banach spaces of measurable functions on {X}, and {\|\cdot\|_i} the corresponding norms. Given {\phi_i\in B_i} and {n\in {\mathbb N}}, the corresponding {n}th correlation is

\displaystyle C_n(\phi_1,\phi_2) := \int \phi_1(x) \phi_2(f^nx) \,d\mu(x) - \int \phi_1\,d\mu \int\phi_2\,d\mu.

We say that {(X,f,\mu)} has exponential decay of correlations with respect to observables in {B_1,B_2} if for any {\phi_i\in B_i}, we have

\displaystyle \limsup_{n\rightarrow\infty} \frac 1n \log |C_n(\phi_1,\phi_2)| < 0. \ \ \ \ \ (1)

 

Equivalently, for every {\phi_1\in B_1} and {\phi_2\in B_2}, there are {\lambda=\lambda(\phi_1,\phi_2)>0} and {K=K(\phi_1,\phi_2)>0} such that

\displaystyle |C_n(\phi_1,\phi_2)| \leq K(\phi_1,\phi_2) e^{-\lambda(\phi_1,\phi_2) n} \text{ for all } n. \ \ \ \ \ (2)

 

One sometimes sees the statement of exponential decay given in the following form, which is formally stronger than (2): there are constants {L>0} and {\lambda>0}, independent of {\phi_1,\phi_2}, such that

\displaystyle |C_n(\phi_1,\phi_2)| \leq L\|\phi_1\|_1 \|\phi_2\|_2 e^{-\lambda n} \text{ for all } n,\phi_1,\phi_2. \ \ \ \ \ (3)

 

In fact, using the Baire category theorem, one can prove that (2) implies (3) under a mild condition on the Banach spaces {B_i}; this is the goal of this post, to show that {\lambda} can be chosen uniformly over all {\phi_i}, and that {K} can be chosen to have the form {K(\phi_1,\phi_2) = L \|\phi_1\|_1\|\phi_2\|_2}. This seems like the sort of thing which is likely known to experts, but I am not aware of the reference in the literature. (I would be happy to learn a reference!)

Proposition 1 Let {(X,f,\mu)} be a probability measure-preserving transformation and {B_1,B_2} Banach spaces of measurable functions on {X} with the following properties:

  • given any {\phi_i\in B_i}, we have {\phi_i\in L^1(\mu)} and {\phi_1\phi_2\in L^1(\mu)};
  • the inclusions {(B_i,\|\cdot\|_i) \rightarrow (L^1,\|\cdot\|_{L^1})} are continuous;
  • for every {\phi_1\in B_1}, the map {(B_2,\|\cdot\|_2) \rightarrow (L^1,\|\cdot\|_{L^1})} given by {\phi_2 \mapsto \phi_1\phi_2} is continuous, and similarly for the map {\phi_1\mapsto \phi_1\phi_2} when {\phi_2} is fixed;
  • the map {f_* \colon B_2 \rightarrow B_2} given by {f_* \phi = \phi\circ f} is bounded w.r.t. {\|\cdot\|_2}.

Under these assumptions, if {(X,f,\mu)} has exponential decay of correlations w.r.t. observables in {B_1,B_2} in the sense of (2), then it also satisfies (3).

To prove the proposition, start by fixing {\phi_1\in B_1} and {\lambda>0}, and consider the function {K_\lambda^{\phi_1} \colon B_2 \rightarrow [0,\infty]} given by

\displaystyle K_\lambda^{\phi_1}(\phi_2) = \sup_n |C_n(\phi_1,\phi_2)| e^{\lambda n}.

Notice that for each {n}, the correlation function {C_n} is bilinear in {\phi_1,\phi_2}, and thus for every {\phi_2,\psi_2\in B_2} and {a,b\in {\mathbb R}}, we have

\displaystyle \begin{aligned} K_\lambda^{\phi_1}(a\phi_2 + b\psi_2) &\leq \sup_n |aC_n(\phi_1,\phi_2)|e^{\lambda n} + |bC_n(\phi_1,\psi_2)|e^{\lambda n} \\ &\leq |a| K_\lambda^{\phi_1}(\phi_2) + |b| K_\lambda^{\phi_1}(\psi_2). \end{aligned} \ \ \ \ \ (4)

 

Consider the following subsets of {B_2}:

\displaystyle V_{\lambda,K}^{\phi_1}(K) := \{\phi_2\in B_2 : K_\lambda^{\phi_1}(\phi_2) \leq K \}.

It follows from (2) that

\displaystyle \bigcup_{\lambda,K>0} V_{\lambda,K}^{\phi_1} = B_2. \ \ \ \ \ (5)

 

Moreover, the sets {V_{\lambda,K}^{\phi_1}} are nested (smaller {\lambda} gives a bigger set, larger {K} gives a bigger set) and so it suffices to take the union over rational values of {\lambda,K}, meaning that we can treat (5) as a countable union. In particular, by the Baire category theorem there are {\lambda,K>0} such that the closure of {V_{\lambda,K}^{\phi_1}} has non-empty interior. The next step is to show that

  • {V_{\lambda,K}^{\phi_1}} is closed, so it itself has non-empty interior;
  • in fact, {V_{\lambda,K}^{\phi_1}} contains a neighbourhood of the origin.

For the first of these, observe that by the assumptions we placed on the Banach spaces {B_1,B_2}, there is a constant {Q=Q(\phi_1)} such that

\displaystyle \|\phi_2\|_{L_1} \leq Q \|\phi_2\|_2 \text{ and } \|\phi_1 \phi_2\|_{L^1} \leq Q(\phi_1) \|\phi_2\|_2

for every {\phi_2\in B_2}. In particular,

\displaystyle \begin{aligned} |C_n(\phi_1,\phi_2)| &\leq Q(\phi_1) \|\phi_2\circ f^n\|_2 + Q\|\phi_1\|_{L^1} \|\phi_2\|_2 \\ &\leq (Q(\phi_1) \|f_*\|^n + Q\|\phi_1\|_{L^1}) \|\phi_2\|_2. \end{aligned}

Given a sequence {(\phi_2^{(k)})_k \subset V_{\lambda,K}^{\phi_1}} such that {\phi_2^{(k)} \rightarrow \phi_2 \in B_2} w.r.t. {\|\cdot\|_2} as {k\rightarrow\infty}, it follows that

\displaystyle |C_n(\phi_1,\phi_2)| \leq \limsup_{k\rightarrow\infty} |C_n(\phi_1,\phi_2^{(k)})| \leq K e^{-\lambda n},

and we conclude that {\phi_2\in V_{\lambda,K}^{\phi_1}}, so this set is closed. In particular, there is {\phi_2\in V_{\lambda,K}^{\phi_1}} and {\varepsilon>0} such that if {\|\psi_2\|_{B_2} \leq \varepsilon}, then {\phi_2 + \psi_2\in V_{\lambda,K}^{\phi_1}}. By the same token {\phi_2 - \psi_2\in V_{\lambda,K}^{\phi_1}}, and now the sublinearity property (4) gives

\displaystyle K_\lambda^{\phi_1}(\psi_2) \leq \tfrac 12 \big(K_\lambda^{\phi_1}(\phi_2 + \psi_2) - K_\lambda^{\phi_1}(\phi_2 - \psi_2)\big) \leq K,

and so {\psi_2\in V_{K,\lambda}^{\phi_1}}. This shows that {V_{K,\lambda}^{\phi_1}} contains a neighbourhood of 0, and writing {L = K/\varepsilon}, we see that for every {\psi_2\in B_2} we have {\varepsilon\frac{\psi_2}{\|\psi_2\|_2} \in V_{K,\lambda}^{\phi_1}}, and so

\displaystyle K_\lambda^{\phi_1}(\psi_2) = \varepsilon^{-1} \|\psi_2\|_2 K_\lambda^{\phi_1} \Big(\varepsilon\frac{\psi_2}{\|\psi_2\|_2}\Big) \leq \varepsilon^{-1}\|\psi_2\|_2 K = L \|\psi_2\|_2.

Thus we conclude that

\displaystyle |C_n(\phi_1,\phi_2)| \leq L(\phi_1) \|\phi_2\|_2 e^{-\lambda(\phi_1)n} \text{ for all } n.

To complete the proof of the proposition, it suffices to apply the same argument once more. Writing

\displaystyle W_{\lambda,K} = \{ \phi_1 \in B_1 : \lambda(\phi_1) \geq \lambda \text{ and } L(\phi_1) \leq L\}

we see from (2) that {B_1 = \bigcup_{\lambda,K>0} W_{\lambda,K}}, and so once again there are {\lambda,K>0} such that the closure of {W_{\lambda,K}} has non-empty interior. Given a sequence {\phi_1^{(k)} \in W_{\lambda,K}} with {\|\phi_1^{(k)} - \phi_1\|_1 \rightarrow 0}, we have {C_n(\phi_1^{(k)},\phi_2) \rightarrow C_n(\phi_1,\phi_2)} for all {n} and all {\phi_2}, and so {\phi_1\in W_{\lambda,K}}, demonstrating that this set is closed.

Thus there are {\phi_1\in W_{\lambda,K}} and {\varepsilon>0} such that {\|\psi_1\|\leq \varepsilon} implies {\phi_1+\psi_1\in W_{\lambda,K}}. In particular this gives {\phi_1 \pm \psi_1\in W_{\lambda,K}}, and so for every {n} and every {\phi_2} we have

\displaystyle \begin{aligned} |C_n(\psi_1,\phi_2)| &= \tfrac 12 |C_n(\phi_1 + \psi_1,\phi_2) - C_n(\phi_1 - \psi_1,\phi_2)| \\ & \leq \tfrac 12 |C_n(\phi_1 + \psi_1,\phi_2)| + \tfrac 12|C_n(\phi_1 - \psi_1,\phi_2)| \\ &\leq K \|\phi_2\|_2 e^{-\lambda n}. \end{aligned}

But then for every {\phi_1\in B_1} we can consider {\psi_1 = \varepsilon \phi_1 / \|\phi_1\|_1}, which has {\|\psi_1\|_1 = \varepsilon}, and so

\displaystyle |C_n(\phi_1,\phi_2)| = \|\phi_1\|_1 \varepsilon^{-1} |C_n(\psi_1,\phi_2)| \leq \|\phi_1\|_1 \varepsilon^{-1} K \|\phi_2\|_2 e^{-\lambda n},

which proves (3) and the proposition.

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About Vaughn Climenhaga

I'm an assistant professor of mathematics at the University of Houston. I'm interested in dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, multifractal analysis, non-uniform hyperbolicity, and things along those lines.
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