  Let $F$ be a $C^3$ 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 shadowing lemma. This
work removes an uncheckable condition of Silnikov [Equation (11), page 625 of
L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968), 624-628]. Also, the
result of Silnikov does not imply Bernoulli shift dynamics of a single map,
rather only provides a labeling of all invariant tubes around the homoclinic
tube. The work of Silnikov was done in ${\mathbb R}^n$, and the current work is
done in a Banach space.
