Things that we will prove:
1. Conformal homeomorphisms all have topological entropy = 0.
2. Pseudo-Anosov surface homeomorphisms are entropy minimal in their homotopy class (we’ll compute the entropy!)
3. By Thurston’s classification of surface diffeomorphisms, after cutting along a system of nonisotopic stabilized curves any surface diffeomorphism can be broken into pseudo-Anosov and conformal pieces (after isotopy). This representative is entropy minimal and we’ll compute all of its ergodic measures.
Along the way, we’ll work out nice upper bounds for the entropy of quasi-conformal maps and wave in passing at the formula for the entropy of holomorphic maps between Kahler manifolds (which can be computed homologically).
Definitions of topological entropy and quasi-conformality will be reviewed and then illustrated with examples. The main ideas of proofs will be sketched through example. In sum, the things we discuss will be beautiful to contemplate.