Topologically Conjugate Classifications Of The Holomorphic Isomorphisms On Simple Connected Domains

In this paper,we are interested in the topologically conjugate classifications of the holomorphic isomorphisms on simple connected domains.We said that two transformations f :X ? X and g : Y ? Y are topologically conjugate if there exists a homeomorphism h : X ? Y such that h (?) f = g (?) h,where (?) is the composition of mappings.Simple connected domains includes : complex plane,extended complex plane,unit disk and upper half plane.For the problem of topologically conjugate classifications of the holomorphic isomorphisms on complex plane,Budnitska gave more general result.He obtained the topologically conjugate classifications of affine operators on finite dimensional vector space.Specifically speaking if the affine operator f(x)= Ax + b has a fixed point,then f is topologically conjugate to its linear part A.If the affine operator f : U ? U has no fixed point,f is topologically conjugate to an affine operator g : U ? U,where U is an orthogonal direct sum of g-invariant subapaces V and W,the restriction g|V is an affine operator with the form(x1,x2,...,xn)?(x1+1,x2,...,xn-1,?xn)under the orthogonal basis of V,? = ±1,and it is uniquely determined by f.The restriction g|W is a linear operator with a nilpotent Jordon form under the orthogonal basis of W,and it is uniquely determined by f.For the problem of topologically conjugate classifications of the holomorphic isomorphisms on extended complex plane,Rybalkina and Sergeichuk have given the answer.It is not known of the completely classifications for the holomorphic isomorphisms on unit disk or upper half plane before.Now we will solve the problem.By rotation theory and some constructions of homeomorphisms,we prove that all holomorphic automorphisms on upper half plane(or unit disk)having no fixed points are topologically conjugate;two holomorphic automorphisms f and g having fixed points are topologically conjugate if and only if?(f)= ±?(g),modZ;a holomorphic automorphism with no fixed points and a holomorphic automorphism with fixed points are not topologically conjugate.