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

We study the ratio εc(ω)/exp(−ηB(ω)), where εc(ω) is the breakdown threshold function for an analytic invariant torus, η is a parameter and B(ω) is the Bruno function, which is purely arithmetic (i.e., it only depends on the number theory properties of ω). We consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with Diophantine rotation frequency. Our numerical experiments, together with some heuristic considerations, strongly suggest that εc(ω)/exp(−ηB(ω)) is not a continuous function on Diophantine numbers ω, for all values of η.