# 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, $w\in L^{\infty}$?

As you are perhaps aware, the standard method for investigating decay of correlations of expanding maps is using operator theory, either directly or via an induced dynamical system. The decay of correlations can be deduced from the spectral properties of the Ruelle transfer operator acting on the space of Hoelder functions, possibly in combination with operator renewal theory if the expansion is not uniform. In your case this would require applying an arbitrary sequence of distinct transfer operators, and knowing the spectral properties of these operators would not be of immediate help.

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 $v$.

You might find interesting to see the generalization of decay of correlation for non autonomous dynamical systems considered in these papers:

Ott, Stenlund and Young, Memory loss for time dependent dynamical systems, Math. Res. Lett 16 (2009)-3, 463-475. http://arxiv.org/abs/1210.0109

Conze and Raugi, Limit theorems for sequential expanding dynamical systems of [0,1] , Contemporary Mathematics, vol. 430 (2007) p. 89-121