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 $A$.