In order to describe the dynamics of a diffeomorphism, one first decomposes the system into invariant elementary pieces, that may be studied separately. For hyperbolic dynamics, the decomposition is provided by Smale's 《spectral decomposition theorem》, and the number of pieces is finite. This talk deals with the decomposition of general smooth diffeomorphisms on surfaces (several notions of pieces are natural).

One will mainly focus on systems which have infinitely many pieces: typical examples have been constructed by Newhouse. I will discuss how the dynamics degenerates along sequences of distinct pieces, in terms of their dynamical invariants (the entropy and the Lyapunov exponents).