Let F be a C3 diffeomorphism on a Banach space B. F has a homoclinic tube asymptotic to an invariant manifold. Around the
homoclinic tube, Bernoulli shift dynamics of submanifolds is
established through a shadowing lemma. This work removes an
uncheckable condition of Silnikov (1968). Also, the result of Silnikov does not imply
Bernoulli shift dynamics of a single map, but rather only provides a
labeling of all invariant tubes around the homoclinic tube. The
work of Silnikov was done in ℝn and the current work is done in a Banach space.