Results and open questions on some invariants measuring the dynamical complexity of a map
Let $f$ be a continuous map on a compact connected Riemannian manifold $M$. There are several ways to measure the dynamical complexity of $f$ and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of $f$, the volume growth of $f$, the rate of growth of periodic points of $f$, some invariants related to exterior powers of the derivative of $f$, and several invariants measuring the topological complexity of $f$: the degree (for the case when the manifold is orientable), the spectral radius of the map induced by $f$ on the homology of $M$, the fundamental-group entropy, the asymptotic Lefschetz number and the asymptotic Nielsen number. In general these relations depend on the smoothness of $f$. Various examples are provided.