Erratum to “Existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near-integrable Hamiltonian systems” [Physica D 114 (1998) 3–80]

As was pointed out by Gallavotti, Gentile and Mastropietro [Physica D 137 (2000) 202–204], Theorem 2.1 of Rudnev and Wiggins [Physica D 114 (1998) 3–80] is false. This theorem claimed an exponentially small estimate (67) for the Fourier coefficients of the splitting distance function (20) of Rudnev and Wiggins (1998). Further, it was used in order to arrive at Theorems 2.2 and 2.3, thus invalidating the proofs of these theorems as given. The present note is contingent upon the new results [M. Rudnev, S. Wiggins, Regular Chaot. Dyn. 4 (1999) 39–58; M. Rudnev, S. Wiggins, Regular Chaot. Dyn. 5 (2000) 227–242] and vindicates the estimate in question. This is achieved via redefining the splitting distance function in accordance with the approach developed in these papers. Such a definition is provided by (11) of this note. Despite the fact that (11) and the definition (20) given by Rudnev and Wiggins (1998) differ only by a near-identity reparameterization, the former affords the estimate (13) of this note which implies (67) of Rudnev and Wiggins (1998). Theorem B of this note gives the correct statement of Theorem 2.1 of Rudnev and Wiggins (1998). Hence, Theorems 2.2 and 2.3 therein stand as they are, modulo a few modifications compiled at the end of this note.