Let $M$ be a smooth manifold and $f\colon M\rightarrow M$. I am interested in the characteristics of $f$ viewed as a discrete dynamical system on $M$. That is, stability, periodicity, Lyapunov theory, etc of the iterates of $f$. Most of the information I can find are specific maps meant to be examples of chaotic dynamics. I'd like to know if there has been any systematic study of the general case.
References to books and papers are welcome.
Although I'm mostly interested in a general theory, I'm also interested in mappings on specific manifolds, including
1) mappings of the circle, sphere, $n$-sphere.
2) mappings of tori
3) mappings of regular surfaces
4) mappings of Lie groups, especially $SU(2)$ and $SO(3)$,