Difference between revisions of "Hyperbolic fixpoint"
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| − | A hyperbolic fixpoint is a [[fixpoint]] $ | + | A hyperbolic fixpoint is a [[fixpoint]] $a$ of $f$ such that $|f'(a)|\neq 0,1$. |
| − | For a locally analytic function with hyperbolic fixpoint at $0$, i.e. $f(z)=c_1 z + c_2z^2$, $|c_1|\neq 0,1$, there is always a locally analytic and injective function $\sigma$ that satisfies the [[Schröder equation]] | + | For a locally analytic function with hyperbolic fixpoint at $0$, i.e. $f(z)=c_1 z + c_2z^2 + \dots$, $|c_1|\neq 0,1$, there is always a locally analytic and injective function $\sigma$ that satisfies the [[Schröder equation]] |
$$\sigma(f(z))=c_1 \sigma(z))$$ | $$\sigma(f(z))=c_1 \sigma(z))$$ | ||
$\sigma$ is unique up to a multiplicative constant and is called the [[Schröder coordinates]] of $f$ at 0. | $\sigma$ is unique up to a multiplicative constant and is called the [[Schröder coordinates]] of $f$ at 0. | ||
Latest revision as of 18:11, 9 June 2011
A hyperbolic fixpoint is a fixpoint $a$ of $f$ such that $|f'(a)|\neq 0,1$.
For a locally analytic function with hyperbolic fixpoint at $0$, i.e. $f(z)=c_1 z + c_2z^2 + \dots$, $|c_1|\neq 0,1$, there is always a locally analytic and injective function $\sigma$ that satisfies the Schröder equation $$\sigma(f(z))=c_1 \sigma(z))$$ $\sigma$ is unique up to a multiplicative constant and is called the Schröder coordinates of $f$ at 0.