Symplectic Dynamics Working Group

One of the most useful tools in quantifying the complexityof a dynamical system are the concepts of topological and metricentropy. While relevant in several contexts, entropy plays aparticularly important role in describing the behaviour of dynamicalsystems in dimension 2. For instance, a famous result by Katok showsthat, for sufficiently smooth systems, the presence of strictlypositive topological entropy implies that there exists a very goodmodel for the dynamics. It is shown that, in particular, if thetopological entropy is at least ln(2), then the complexity observed isat least as large as that of the classical Bernoulli system (that is,it is at least as complex as the process of flipping a honest coininfinitely many times).

On the other hand, one can consider if there exists any sort ofdescription of the dynamics when the entropy is null. In recent workswith P. Le Calvez we have shown that, at least for planarhomeomorphisms, a lot of information can be derived. In particular,one can show that in many ways the phase portrait share many properties with thatof a flow. Furthermore, conservative systems with zero entropy behavesimilarly to integrable systems.

Joint work w. P. Le Calvez.