Invariant curves of some discrete dynamical systems

The classical problem on construction of continuous iterations of an analytical map is considered as a problem on construction of invariant curves of discrete dynamical systems. Such systems are often studied as reductions of continuous dynamical systems (Poincarè map). The existence of analytical invariant curves in a discrete dynamical system implies (locally) the existence of an additional analytical first integral in the continuous dynamical system. However, the proofs of existence of such integrals are extremely rare, since these proofs are usually based on convergence of formal power series representing these curves. We give some examples of discrete dynamical systems invariant curves of which are given by a fortiori divergent series but are analytical nonetheless. In particular, we give an example of an integrable discrete dynamical system which has chaotic trajectories.