A critical function for an area-preserving map associates with each fixed irrational rotation number omega the breakdown threshold K( omega ) of the corresponding KAM invariant circle. Understanding the structure of such a function and obtaining good estimates and approximations to it is a problem of fundamental theoretical importance and also has relevance to many applications. The authors present strong numerical evidence that a purely arithmetic function, the Brjuno function B( omega ), which only depends on the nearest-integer continued fraction expansion of the rotation number, omega , provides a good approximation to log(K( omega )) for the standard map, which is one of the most commonly studied area-preserving maps. They discuss the relationship of the Brjuno function to critical functions in other small-divisor problems, remark on the relevance of their results in explaining the modular smoothing technique of Buric and Percival (1991) and prove that K( omega )>0 for all complex omega with a non-zero imaginary part.
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