Analytic singularities near radial points

In this thesis, we applied tools of algebraic analysis and knowledge of symplectic geometry and contact geometry to give a normal form of certain class of microdifferential operators, and then studied analytic singularities of solutions with the advantage of normal form. The microdifferential operators are of real analytic coefficients, real principal symbols and simple characteristics near radial points. We linearized the contact vector fields with real analytic coefficients, classified the radial points and find an exact normal form of our operators. In the last two chapters, by restricting our discussion in the space of Fourier hyperfunctions, first we fully studied the analytic singularities in two dimensional case, and gave some estimates of singularities in higher dimensional cases. Roughly, near an attracting (resp. repelling) generic radial point, we can found solutions with minimal analytic singularity, i.e. the radial direction. Furthermore, near a non-attracting (resp. non-repelling) radial point, if the radial direction is contained in analytic wavefront set of the solution, then either the intersection of analytic wavefront set with stable manifold or with unstable manifold is non-empty. Moreover we discussed solution with prescribed singularities and gave a description of propagation of analytic singularities, especially in three dimensional case.