THE CRITICAL FUNCTION FOR THE SEMISTANDARD MAP

For the semistandard map F(x, y) = (x + y + ie(ix), y + ie(ix)), we consider the critical function K(ss)(omega), defined as the radius of convergence of a series expansion of a complex invariant curve of rotation number omega, and show that log K(ss)(omega) + 2SIGMAq(k)-1 log q(k+1) is bounded on the set of omega where it is well-defined, where {q(k)} are the denominators of the convergents to the real number omega. We discuss the implications for critical functions for the standard map.