ON THE STANDARD MAP CRITICAL FUNCTION

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. We 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. We discuss the relationship of the Brjuno function to critical functions in other small-divisor problems, remark on the relevance of our results in explaining the modular smoothing technique of Buric and Percival and prove that K(omega) > 0 for all complex-omega with a non-zero imaginary part.