Hyperbolic fixpoint

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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.