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 )+2 Sigma q_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.