Theorem: every curve interval bounded by a secant line has its midpoint in the projection of where.

With this theorem the is verified among other things the Taylor’s theorem.

]]>Some related results are given by Rodrigues and Sola-Morales in “Known results and open problems on linearization in Banach Spaces”, Sao Paulo J. Math. Sci. 6, 2(2012), 375-384.

]]>Well, another obvious obstruction would be that would have to be injective.

If one requires all maps to be holomorphic, then any global linearising map must be an analytic continuation of a local linearising map , which one can show to be unique once one normalises . So it’s basically a question of how far one can analytically continue and keep it invertible.

]]>Consider the possibility of linearizing globally over its whole domain in the attractive case where is simply connected:

A necessary condition seems to be that is the only(!) fixed point for in U. Is this the only obstruction for the global linearization of over ? ]]>

This is true; however, in my post I wish to emphasise that the precise value of C is not terribly important for this type of argument, and can change in different instances of that argument (e.g. in Lemma 4, the requirement on C will be a little different).

]]>I don’t know of any work in this direction, but there is some literature on p-adic versions of Siegel’s linearisation theorem, in which the complex plane is replaced by the p-adic completion of .

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