Minimality And Minimal Decomposition Of P-adic Rational Maps With Potential Good Reductions

Let p be a prime number and let Q_p be the field of p-adic numbers.A rational mapψ ∈ Q_p(t)induces a continuous map on the projective line P~1(Q_p)over Q_p.We study the dynamical behavior of the system(P~1(Q_p),ψ)in two aspects:the minimality and minimal decomposition of rational maps with potential good reductions;the complexity of the Julia sets of a family of rational maps.The rational maps with potential good reduction are classified in two cases:ramified potential 1-Lipschitz continuous and unramified potential 1-Lipschitz continuous.This thesis is divided into three parts.In the first part,the rational maps ψ∈ Q_p(t)with potential good reductions are studied as dynamical systems on P~1(Q_p).The dynamical behavior of these maps are entirely described by their minimal decompositions.We give the criterion of minimality for the dynamical systems of rational maps with potential good reduction.For any prime number p,a condition in terms of the coefficients the rational map is proved to be necessary for the map being minimal,having potential good reduction and ramified potential 1-Lipschitz continuous,sufficient for the map being minimal and ramified potential 1Lipschitz continuous.In the second part,we construct a rational map ψ∈Q_p(t)which is ramified potential 1-Lipschitz,but does not have potential good reduction,and the system(P~1(Q_p),ψ)is minimal.In the third part,on any finite algebraic extension L of the field Q_p,it is proved that there exist rational maps ψ∈L(t)such that the dynamical system(P~1(L),ψ)has empty Fatou set,i.e.the iteration family {ψn:n≥ 0} is nowhere equi-continuous.