On the relationship between the Bruno function and the breakdown of invariant tori

We study the ratio epsilon(c)(omega)/exp(-eta B(omega)), where epsilon(c)(o mega) is the breakdown threshold function for an analytic invariant torus, eta is a parameter and B(omega) is the Bruno function, which is purely arit hmetic (i.e., it only depends on the number theory properties of omega). We consider the standard map as a model and we focus our analysis on the expo nential decay of the chaotic regions close to an invariant torus with Dioph antine rotation frequency. Our numerical experiments, together with some he uristic considerations, strongly suggest that epsilon(c)(omega)/exp(-eta B( omega)) is not a continuous function on Diophantine numbers w, for all valu es of eta. (C) 2000 Published by Elsevier Science B.V. All rights reserved.