Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables

Abstract:

This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in $\mathbb {R},\mathbb {C},$ or ${\mathbb {R}^2}$), (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, $k$th 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 $z \mapsto z + {z^2}$ should be better thought of as the map $(x,y) \to (x + {x^2} - {y^2},y + 2xy)$.