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Extra resources for A Course of Pure Geometry

Example text

2. Let F be a quasi-parabolic germ of diffeomorphism of C2 at 0. Then ν(F) = +∞ if and only if there exists a germ of (holomorphic) curve through 0 that consists of Àxed points of F. 3. Let F be a quasi-parabolic germ of diffeomorphism of C2 at 0. If F is dynamically separating then there exist ν(F) − 1 parabolic curves for F at 0. The argument in [10] is based on a series of blow-ups and changes of coordinates which allow to write F into a suitable form so that one can write a similar operator to the one deÀned by Hakim and prove that its Àxed points in a certain Banach space of curves are exactly the sought parabolic curves.

In the same direction, in the recent paper [6], Abate and Tovena studied real dynamics of complex homogeneous vector Àelds. Besides its intrinsic interest, this is an useful problem to study because the discrete dynamics of the time 1-map is encoded in the real integral curves of the vector Àeld, and time 1-maps of homogeneous vector Àelds are prototypical examples of holomorphic maps tangent to the identity at the origin. The main idea here is that, roughly speaking, integral curves for homogeneous vector Àelds are geodesics for a meromorphic connection on a projective space.

2) ( ∈ {0, 1}). Summation starts at = 0 unless F(0) ∈ {0, ∞}, in which case it starts at = 1. 1 The two driving functions F 1 This choice is to ensure near-invariance under the change F(x) → 1/F(1 − x). 5. 38 Jean Ecalle and Shweta Sharma and f are connected under F ≡ exp(− f ). e. when f is holomorphic. ). As for the above deÀnition, it is less arbitrary than may seem at Àrst sight. Indeed, none of the following changes: (i) changing the grid {k/n} to {Const k/n} (ii) changing the lower summation bounds from 0 or 1 to 2,3 .

by Jason
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