ENTIRE-FUNCTIONS, IN THE CLASSIFICATION OF DIFFERENTIABLE GERMS TANGENT TO THE IDENTITY, IN ONE OR 2 VARIABLES

This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in R, C, or R(2)), (maps f defined near 0, such that f(0) = 0, and f'(0) is the identity). Proofs are mostly original. The paper is mostly-oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, kth root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition 2 the function z --> z + z(2) should be better thought of as the map(x,y) --> (x + x(2) - y(2), y + 2xy).