Difference between revisions of "Super-attracting fixpoint"
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A fixpoint $a$ of $f$ such that $f'(a)=0$. | A fixpoint $a$ of $f$ such that $f'(a)=0$. | ||
| − | For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the [[Böttcher equation]] $\beta(f(z))=\beta( | + | For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the [[Böttcher equation]] |
| + | $$\beta(f(z))=\beta(z)^p,$$ | ||
| + | with $\beta(0)=0$, $\beta'(0)\neq 0$ is satisfied. $\beta$ is unique up to taking positive integer powers and multiplication by ($p-1$)-root of unity. It is called the [[Böttcher coordinate]]. | ||
Latest revision as of 12:47, 7 June 2011
A fixpoint $a$ of $f$ such that $f'(a)=0$.
For an at $0$ analytic function $f(z)=c_p z^p + c_{p+1}z^{p+1} + \dots$, $p\ge 2$, $c_p\neq 0$, one can always find a locally analytic and injective function $\beta$ such that the Böttcher equation $$\beta(f(z))=\beta(z)^p,$$ with $\beta(0)=0$, $\beta'(0)\neq 0$ is satisfied. $\beta$ is unique up to taking positive integer powers and multiplication by ($p-1$)-root of unity. It is called the Böttcher coordinate.