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Diophantine conditions for the linearization of commuting holomorphic functions

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  • We prove the local simultaneous linearizability of a pair of commuting holomorphic functions at a shared fixed point under a very general - we conjecture optimal - diophantine condition. Let $f,g :\mathbb{C} \to \mathbb{C}$ with a common fixed point at the origin and suppose that $f(z) = \lambda z + \cdots$ and $\lambda \ne 0$. The map, $f,$ is called linearizable if there is an analytic diffeomorphism, $h$, which conjugates $f$ with its linear part in a neighborhood of the origin, i.e., $h^{-1} \circ f \circ h (z) = \lambda z$ where $\lambda = f'(0).$ Two such diffeomorphisms are simultaneously linearizable if they are linearized by the same map, $h$. If $|\lambda| = 1$ then the situation is delicate. Nonlinearizable maps are topologically abundant, i.e., for $\lambda$ in a dense co-meager set in $\mathbb{S}^1$ there exist nonlinearizable analytic maps with linear coefficient $\lambda$. In contrast there is a diophantine condition on $\lambda$ that is satisfied by a set of full measure in $\mathbb{S}^1$ which assures linearizability of the map $f$.
    Mathematics Subject Classification: Primary: 58F23; Secondary: 30D05, 58F36.


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