'Expansion prevails fractioning', is a quote attributed to Ya. Sinai, which describes a key ingredient of the local ergodic theorem and is one of the reasons for his fame. If the local ergodic theorem applies correctly a nonuniformly dynamical system with singularities can be shown to be ergodic. At its core lies the Hopf method. As a motivation we will start illustrating the Hopf method in a simpler setting, namely, a uniformly hyperbolic dynamical system without singularities and explain the transition of this method to nonuniform dynamics with singularities.