Smooth Linearization Of Three-dimensional Hyperbolic Diffeomorphisms

Dynamic systems study the laws of state changing with time under certain rules and most dynamic systems are nonlinear.Linearization is one method to study the local qualitative properties of systems.It can transform nonlinear system problems into its corresponding linear ones,and further study dynamic properties of nonlinear systems through linear systems.Specifically,the linerization of the mapping F means that there is a homeomorphismΦ)satisfying the conjugate equation Φ(?)F=S(?)Φ,where S is the linear part of F at its fixed point.In the process of linearization,C1 linearization preserves smooth dynamical behaviors and distinguishes qualitative properties in the characteristic direction.Hence,based on the known results,we will make a further investigation of C1 linearization for hyperbolic diffeomorphisms in this thesis.The problem is further divided into Poincaré domain case and Siegel domain case in according to the positions of the eigenvalues for the given mapping at the hyperbolic fixed point.In Chapter 2,we study the linearization problem in Poincare domain.Previous researchers considered the C1 linearization of C1,α differential homeomorphism on R2 and the C1 linearization of C1,α differential homeomorphism on Rn under the condition that the coordinate surfaces are invariant.In this thesis,we study the C1 linearization problem of C1,α differential homeomorphism on R3 without the invariance of the coordinate surfaces.In order to ensure the smoothness of conjugate mapping,the key to the proof is to effectively control the iteration of given mapping,overcame by finding the invariant surface and invariant curve on it.The increase in dimensionality makes it more difficult to find invariant surface and invariant curves.In Chapter 3,we study the linearization problem in Siegel domain.The Siegel domain case is more complex than that in Poincaré domain.This is because that there are divergence factors inevitably no matter considering the given mapping F or its inverse F-1.On the basis of previous studies on the linearization of C1,α differential homeomorphism of C1,β on R2,we further extend the results to ones in Rn.The increase of dimension not only makes the form of conjugate mapping more complex,but also makes its convergence more difficult to guarantee.By using the function sequence with the iteration of the given mapping,the solution of the conjugate equation is approximated.By verifying the uniform convergence of the series of function terms corresponding to the function sequence,it is proved that this mapping can be locally linearized by C1,β.