We study invariant measures of families of monotone twist maps
S
γ
(
q
,
p
)
S_{\gamma }(q,p)
=
=
(
2
q
−
p
+
γ
⋅
V
′
(
q
)
,
q
)
(2q-p+ \gamma \cdot V’(q),q)
with periodic Morse potential
V
V
. We prove that there exist a constant
C
=
C
(
V
)
C=C(V)
such that the topological entropy satisfies
h
t
o
p
(
S
γ
)
≥
log
(
C
⋅
γ
)
/
3
h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3
. In particular,
h
t
o
p
(
S
γ
)
→
∞
h_{top}(S_{\gamma }) \to \infty
for
|
γ
|
→
∞
|\gamma | \to \infty
. We show also that there exist arbitrary large
γ
\gamma
such that
S
γ
S_{\gamma }
has nonuniformly hyperbolic invariant measures
μ
γ
\mu _{\gamma }
with positive metric entropy. For large
γ
\gamma
, the measures
μ
γ
\mu _{\gamma }
are hyperbolic and, for a class of potentials which includes
V
(
q
)
=
sin
(
q
)
V(q)=\sin (q)
, the Lyapunov exponent of the map
S
S
with invariant measure
μ
γ
\mu _{\gamma }
grows monotonically with
γ
\gamma
.