A set $G$ endowed with an associative binary operation is called a semigroup if it possesses an identity element. Thus a semigroup is short of a group in that it may not be closed under inverses.
Let $X$ be a compact metric space and $V$ be a finite dimensional real vector space.
In Random Walks on Reductive Groups by Benoist and Quint, the following definition is found on pg 42 (Section 3.3.2)
Definition. A continuous function $\sigma:G\times X\to V$ is called a cocyle if $$\sigma(gg', x) = \sigma(g, g'x)+ \sigma(g', x)$$ for any $g, g'\in G$ and $x\in X$.
This is an abstract definition but the text gives no examples to illustrate this via examples. Can someone provide some examples to motivate the concept? Thanks.
Consider the group of affine transformation of $\mathbb R^n$, the linear part is $\mathrm {GL}_n\mathbb R$ while the translation part lives in $\mathbb R^n$. We thus have a decomposition $$\begin{matrix} \mathrm{Aff}(\mathbb R^n) &\rightarrow& \mathrm {GL}_n\mathbb R \times \mathbb R^n \\ \phi&\mapsto& \phi_L,\tau\end{matrix}$$
Let $\rho : G \rightarrow \mathrm{Aff}(\mathbb R^n)$ be an affine representation of a group $G$. Let us write $\rho_L : G\rightarrow \mathrm{GL}_n{\mathbb R}$ be linear part of this morphism and write $\tau : G\rightarrow \mathbb R^n$ be the translation part of this morphism. Notice that while $\rho_L$ is a group morphism, $\tau$ is a cocyle for the action of $G$ on $\mathbb R^n$ given by $\rho_L$.