Quantitative estimates for periodic points of reversible and symplectic holomorphic mappings

Abstract.

We show that reversible holomorphic mappings of ${\bf C}^2$ have periodic points accumulating at an elliptic fixed point of general type. On the contrary, we also show the existence of holomorphic symplectic mappings that have no periodic points of certain periods in a sequence of deleted balls about an elliptic fixed point of general type. The radii of the balls are carefully chosen in terms of the periods, which allows us to show the existence of holomorphic mappi ngs of ${\bf C}^2$ that are not reversible with respect to any $C^1$ involution with a holomorphic linear part, and that admit no invariant totally real and $C^1$ real surfaces.