Finite Determinacy And Polynomial Normal Forms For Diffeomorphisms Near Some1-resonant Fixed Points

This present thesis primarily consists of two parts: In the first part, we discuss thefinite determinacy and polynomial normal forms of quasi-strongly1-resonant diffeo-morphisms near the fixed point on R3. In the second part, we discuss the finite de-terminacy and polynomial normal forms of strongly1-resonant diffeomorphisms nearthe fixed point on Rn. This thesis based on Poincare-Dulac normal form theory andBelistkii theory, use the classical normal form method and reference for the methodof the1-resonant vector field to study the normal forms of these1-resonant diffeomor-phisms.In the first part, we discuss the finite determinacy and polynomial normal forms ofquasi-strongly1-resonant diffeomorphisms near the fixed point on R3. For the quasi-strongly1-resonant diffeomorphisms, we discuss its4kinds of eigenvalues respectively,then use resonant transformation and Belistkii theory to obtain the polynomial normalforms and discuss their finite determinacy in all4case.In the second part, we discuss the finite determinacy and polynomial normal formsof strongly1-resonant diffeomorphisms near the fixed point on Rn. We lay out thegenericity condition and degenerated condition of strongly1-resonant diffeomorphisms.We use resonant transformation to obtain the polynomial normal forms and discuss theirfinite determinacy under this two conditions.