# Decay of Correlation, references for a non-standard way

In ergodic theory is common to use the decay of correlation property to deduce properties analogues to those of i.i.d. random variables.

Call $X\doteq [0,1].$

Examples of decay of correlation properties in dynamics:

1) For $f:X\to X,$ $f(x)=Nx$ mod1, where $N\in \{2,3,4,\ldots,\}$ is fixed, the Lebesgue measure $\lambda$ on $[0,1]$ is an invariant measure for $f$ and for $\epsilon>0$ there is a constant $C$ such that

$\left| \int_X v\cdot w\circ f^n d\lambda-\int_X v d\lambda\cdot \int_X w d\lambda \right|\leq C\cdot \left(\frac{1}{N}+\epsilon\right)^n \cdot\left\Vert v\right\Vert \cdot|w|_{\infty}$

for all $v,w:X\to \mathbb{R},$ $v$ Holder, $w\in L^{\infty}.$

2) For $\alpha\geq 0$ the map

$f:X\to X,$

$f(x)=\begin{cases} x(1-2^{\alpha}x^{\alpha}) & ,x<1/2,\\ 2x-1 & ,x\geq 1/2.\end{cases}$

has a unique finite invariant measure $\mu$ (up to scaling) if and only if $\alpha<1.$ And that in this case, there is a constant $C$ such that

$\left| \int_X v\cdot w\circ f^n d\mu-\int_X v d\mu\cdot \int_X w d\mu \right|\leq C \cdot \frac{1}{n^{1/\alpha-1}} \cdot\left\Vert v\right\Vert \cdot|w|_{\infty}$

for all $v,w:X\to \mathbb{R},$ $v$ Holder, $w\in L^{\infty}.$

A good reference for other examples and further results is the book Positive Transfer Operators and Decay of Correlations, by Viviane Baladi.

My question is where can I look for literature in which people has developed ideas in the way that I explain in what follows.

Suppose that $x=(x_n)_n\in \{2,3,\ldots\}^{\mathbb{N}}$ and define $f_{x_n}:X\to X,$ $f_{x_n}(y)=x_n \cdot y$ mod 1 for all $n\in\mathbb{N}.$

What about the asymptotic properties of

$\left| \int_X v\cdot w\circ f_{x_n} d\lambda-\int_X v d\lambda\cdot \int_X w d\lambda \right|?$

For example:

1)Is it true that for $z=(2,3,2,2,2,3,\ldots)$ an elements of $\{2,3\}^{\mathbb{N}}$ and $x=(x_n)_n$ with $x_n=\prod_{i=1}^n z_i$ we have that for $\epsilon>0$

$\left| \int_X v\cdot w\circ f_{x_n} d\lambda-\int_X v d\lambda\cdot \int_X w d\lambda \right|\leq C \cdot \left(\frac{1}{2}+\epsilon\right)^n\cdot\left\Vert v\right\Vert \cdot|w|_{\infty}$

for all $v,w:X\to \mathbb{R},$ $v$ Holder, $w\in L^{\infty}$?

Since you are interested specifically in the linear expanding maps $x \mapsto nx \mod 1$, I suggest that you argue directly using Fourier analysis. If you restrict $v$ to a high enough smoothness category ($C^k$ for some sufficiently large $k$, or perhaps even real analytic) then you should be able to obtain a correlation estimate in a fairly direct manner by exploiting the rate of decay of the Fourier coefficients of $v$.