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.

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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