Conley-Zehnder index and bifurcation of fixed points of Hamiltonian maps

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley-Zehnder index of a fixed point changes. The main result is that when the Conley-Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.