Equivalences of holomorphic mappings in one and several complex variables

This thesis attempts to address two issues. First, we look at germs of holomorphic maps of the form f(z) = z + amzm +··· defined in a small neighborhood of the origin. We show that such germs may be conjugated with the time-one map F0, p of the vector field V(z) = zp66 z via a homeomorphism H(z) = cz + h(z) where h is a Cinfinity mapping in a punctured neighborhood of the origin and c ∈ C. Hence, this classification agrees with the topological one. Moreover, this mapping h satisfies the same estimates as its topological counterpart. However, in the case that m = 2, we show that any homeomorphism H which is real-analytic with real-analytic inverse conjugating f with the time-one map F0,2 = z1-z must necessarily give rise to holomorphic mapping g which conjugates f with F 0,2.;We then look at holomorphic germs in a neighborhood of the origin in several complex variables. We look at mappings of the form Fz,w=m z+fz,l1w 11+g1z &parr0;,&cdots;,lnwn&parl0;1+g nz , where f(z) = j=minfinity ajzj for j ≥ 2 and gl(z) = k=ninfinity blkzk for k ≥ 1 and l = 1,···,n. In the case that |mu| ≠ 1, we show that any map F of this type may be holomorphically linearized. If mu = 1, we show that the map F may be formally reduced to the form F0z,w =z+zm+nz 2m-1,l1w1 1+g1&d5; z, &cdots;,lnwn 1+g&d5;nz , where g˜i is a holomorphic polynomial of degree at most m - 1. Note that if no resonances are present between the eigenvalues 1, l1,&cdots;,ln , this further reduces the Poincare-Dulac normal form of F. Finally, we demonstrate various equivalences (topological, holomorphic on sectors) of mappings of this type.