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@LoïcTeyssier Any way we just talked about general cases what about a particular (however non injective case $z\mapsto e^{\bar{z}} +z$ it is fixed point free. is it conjugate to a holomorphic map?
@LoïcTeyssier To seek non triviality we may impose the extra condition that the group $G \supset \mathbb{C}\rtimes \mathbb{C}^*$ should not be a product or a semidirect product
@LoïcTeyssier A complex or real Lie group properly contains $\mathbb{C}\rtimes \mathbb{C}^*$ which extend the above mentioned action in an effective manner by real analytic diffeomorphism each conjugate to some bioholomorphism $az+b$. I think this is a new concept
@LoïcTeyssier +1 for your very informative answer. I said the problem is not new but I did not say the problem is solved . I meant generally speaking the problem of analysis around fixed point in the Holomorphic case is an old problem. But here i suggest the case no materials we have, no fixed point no periodic orbit. My precise question: consider the usual action of $\mathbb{C}\rtimes \mathbb{C}^*$ on the plane via $az+b$. Can one extend this action in the following manner:
@Loic but the local version is not a new concept, is not a new problem. On the other hand globally thinking one may pose the following: are there sufficiently high dimensional lie groups of analytic diffeomorphism of the plane consist of diffeomorphisms conjugate to bioholomorphis of C or D? Motivation: The bioholomorphism group is small!
Again with a global point of view, let we have an algebraic foliation of plane whose flow is a complete flow(complete algebraic flow is classified by Chicone and Sotomayor). The flow $\phi_t$ is real analytic . Can one say that it is conjugate to a global linear map?(az+b)
Yes local analysis around fixed points and periodic orbits is a complicated problem. but I wonder why OP uses the Riemann surface terminology (if he is not searching for global analysis?) With a global view point existence of two different period for a diffeomorphism simply implies a negative answer . So my question: if there is no any object to work(no periodic orbit) how can one apply your answer to the main question?
