multiplicative cocycle

Let $f:M\to M$ be a measurable transformation, and let $\mu$ be an invariant probability measure. Consider $A:M\to GL(d,\text{𝐑})$, a measurable transformation, where GL(d,R) is the space of invertible square matrices of size $d$. We define $A^{-1}:M\to GL(d,\text{𝐑})$ by $A^{-1}(x)={[A(x)]}^{-1}$. Then we define the sequence of functions:

$${\varphi }^n(x)=A(f^{n-1}(x))\mathrm{\cdots }A(f(x))A(x)$$

$${\varphi }^{-n}(x)={[{\varphi }^n(f^{-n}(x))]}^{-1}$$

for $n\ge 1$ and $x\in M$.

It is easy to verify that:

$${\varphi }^{m+n}(x)={\varphi }^n(f^m(x)){\varphi }^m(x)$$

for $n,m\in \text{𝐙}$ and $x\in M$.

The sequence ${({\varphi }^n)}_n$ is called a multiplicative cocycle, or just cocycle defined by the transformation $A$.

Title	multiplicative cocycle
Canonical name	MultiplicativeCocycle
Date of creation	2014-03-19 22:13:54
Last modified on	2014-03-19 22:13:54
Owner	Filipe (28191)
Last modified by	Filipe (28191)
Numerical id	4
Author	Filipe (28191)
Entry type	Definition
Synonym	cocycle; multiplicative linear cocycle
Related topic	Furstenberg-Kesten theorem
Defines	multiplicative cocycle