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 $f$.