## 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.