# multiplicative cocycle

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

 $\phi^{n}(x)=A(f^{n-1}(x))\cdots A(f(x))A(x)$
 $\phi^{-n}(x)=[\phi^{n}(f^{-n}(x))]^{-1}$

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

It is easy to verify that:

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

for $n,m\in\textbf{Z}$ and $x\in M$.

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

Title multiplicative cocycle MultiplicativeCocycle 2014-03-19 22:13:54 2014-03-19 22:13:54 Filipe (28191) Filipe (28191) 4 Filipe (28191) Definition cocycle; multiplicative linear cocycle Furstenberg-Kesten theorem multiplicative cocycle