  There is a $C^1$-residual (Baire second class) subset $\mathcal{R}$ of
symplectic diffeomorphisms on $2d$-dimensional manifold, $d\geq 1$, such that
for every non-Anosov $f$ in $\mathcal{R}$ its topological entropy is lower
bounded by the supremum of the Lyapunov exponents of their hyperbolic periodic
points in the \emph{unbreakable central subbundle} (i.e., central direction
with no dominated splitting) of $f$. The previous result deals with the fact
that for $f$ in a residual set $\tilde{\mathcal{R}}$ of symplectic
diffeomorphisms (containing $\mathcal{R}$) satisfies a trichotomy: or $f$ is
Anosov or $f$ is robustly transitive partially hyperbolic with {\em unbreakable
center} of dimension $2m$, $0 < m < d$, or $f$ has totally elliptic periodic
points dense on $M$. In the second case, we also show the existence of a
sequence of $m$-{\em elliptic} periodic points converging to $M$. Indeed,
$\tilde{\mathcal{R}}$ contains an open and dense subset.
