Difference between revisions of "Perturbed Fatou coordinates"
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| − | + | While [[Fatou coordinates]] are the (injective parts of the) [[principal Abel function]]s of a holomorphic function $f$ at a given fixpoint, | |
| + | for simplicity this fixpoint is assumed to be at 0: | ||
| + | $$f(z)= z + c_{m+1} z^{m+1} + o(z^{m+1}),$$ | ||
| + | |||
| + | 'perturbed' Fatou coordinates refer to a function that is 'disturbed' by some small $c_0\in\C$: | ||
| + | $$f(z)=c_0 + z + c_{m+1} z^{m+1} + o(z^{m+1})$$ | ||
| + | |||
| + | This function usually has $2m$ fixpoints in a vicinity of 0. | ||
| + | |||
| + | For the details of the case $m=1$ I refer to [[Shishikura_perturbed_Fatou_coordinates|Shishikuras presentation]]. | ||
Revision as of 11:34, 5 June 2011
While Fatou coordinates are the (injective parts of the) principal Abel functions of a holomorphic function $f$ at a given fixpoint, for simplicity this fixpoint is assumed to be at 0: $$f(z)= z + c_{m+1} z^{m+1} + o(z^{m+1}),$$
'perturbed' Fatou coordinates refer to a function that is 'disturbed' by some small $c_0\in\C$: $$f(z)=c_0 + z + c_{m+1} z^{m+1} + o(z^{m+1})$$
This function usually has $2m$ fixpoints in a vicinity of 0.
For the details of the case $m=1$ I refer to Shishikuras presentation.