Smooth dynamics is concerned with smooth transformations of Riemannian manifolds. It is one of the most exploited areas of dynamical systems, and many papers and books are devoted to this branch. We refer the reader to the book [Katok and Hasselblatt, 1995] as a primary reference. These studies require background in smooth geometry, hyperbolic dynamics, foliation theory, and many more. Also here the entropy is one of the most important subjects.

As this book is designed to be self-contained, and there is obviously no room to provide all that background, deprived of the basic tools, we will actually be able to do very little. In fact we will prove only one rather elementary fact: an estimate of the measure-theoretic entropy in terms of characteristic exponents, a weaker version of the Margulis–Ruelle estimate of entropy for ergodic measures. Besides that, we will only state several results without a proof: the Pesin Entropy Formula, the Buzzi–Yomdin estimate of the topological tail entropy for Cr maps, and some results and questions concerning symbolic extensions.

Margulis–Ruelle Inequality and Pesin Entropy Formula

Let T : M → M be a C1 transformation of a compact Riemannian manifold M of dimension dim. We refrain from providing the detailed definition of the derivative DxT of T at x, which is a transformation defined on the tangent bundle of M.