Several Problems In Dynamical System And Schrondinger Operator

In this thesis,our goal is to study two problems in complex dynamical system and discrete Schrodinger operator.In the first part,we try to find a new geometric characterization of Julia set based on Ahlfors covering surface.The main technique in this part is to use Ahlfors-Shimizu characteristic and Bergweiler,Rippon and Stallard's version of Eremenko point argument.In the second part,we focus our self on the problem of spectrum of Thue-Morse Schrondinger operator.We find three dense subsets ??,??and ?? of the spectrum of the Thue-Morse Hamiltonian,such that each energy in ?? is extended;each energy in ??is pseudo-localized and each energy in ?? is one-sided pseudo-localized.We also obtain exact estimations on the norm of the transfer matrices and the norm of the formal solutions for these energies.Especially,for E ? ??? ??,the norms of the transfer matrices behave like The local dimensions of the spectral measure on these subsets are also studied.The local dimension is 0 for energy in ??and is larger than 1 for energy in ??? ??.In summary,the Thue-Morse Hamiltonian exhibits mixed spectral nature.