Random Dynamical Systems: Intuitive Understanding I am having trouble understanding this definition, [Arnold: Random Dynamical Systems]:
Definition: A measurable random dynamical system on the measurable space $(X,\mathcal{B})$ over a metric dynamical system $(\Omega,\mathcal{F},\mathbb{P},(\theta(t))_{t\in\mathbb{T}})$ as a map $\phi$ with time $\mathbb{T}$ is a mapping
$$\phi:\mathbb{T}\times\Omega\times X\to X,$$
where $\phi$ is measurable and satisfies the cocycle property. $\blacksquare$
I understand that a metric dynamical system is a family of iterative mappings on the  space $(\Omega,\mathcal{F},\mathbb{P})$ that act as endomorphisms, but I don't understand how it all ties together. As far as I can tell, $\phi$ is a deterministic map. 
I don't understand how randomness is described. Can anyone enlighten / inspire me?
 A: For every $(t,\omega,x)$ in $\mathbb T\times\Omega\times X$, $\phi(t,\omega,x)$ is some point in $X$ hence, for every $(t,x)$ in $\mathbb T\times X$, $\phi(t,x):\omega\mapsto\phi(t,\omega,x)$ is a random variable with values in $X$. 
Furthermore, one asks that $\phi(t+s,x)=\phi(t,\phi(s,x))$ for every $(t,s,x)$ in $\mathbb T\times\mathbb T\times X$, in the sense that $\phi(t+s,\omega,x)=\phi(t,\omega,\phi(s,\omega,x))$ for every $(t,s,\omega,x)$ in $\mathbb T\times\mathbb T\times\Omega\times X$. In other words, each $\phi(\ ,\omega,\ )$ is a flow on $\mathbb T\times X$.
Edit: Here is a dynamical system. Assume that $\mathbb T=\mathbb N_0$, $X=\mathbb R$, that $\Omega=\{0,1\}^\mathbb N$ is endowed with its cylindrical sigma-algebra, and consider the shifts $(\theta_n)_{n\geqslant0}$ defined by $\theta_n(\omega)=(\omega_{k+n})_{k\geqslant1}$ for every $n\geqslant0$ and $\omega=(\omega_k)_{k\geqslant1}$ in $\Omega$, for example, $\theta_6(\omega_1,\omega_2,\omega_3,\ldots)=(\omega_7,\omega_8,\omega_9,\ldots)$. Finally, define $\phi(1,\omega,x)=A_{\omega_1}(x)$ for every $x$ in $X$ and every $\omega=(\omega_k)_{k\geqslant1}$ in $\Omega$, where $A_0$ and $A_1$ are some given  functions on $\mathbb R$. In words, $\phi(1,\ ,\ )$ applies $A_0$ or $A_1$ to $x$, the index $0$ or $1$ being chosen uniformly randomly. 
Let me suggest that you check that, for every $n\geqslant0$, 
$$
\phi(n,\omega,x)=A_{\omega_n}\circ \cdots\circ A_{\omega_2}\circ A_{\omega_1}(x),
$$
and that the cocycle property holds in this setting.
