Fix a nontrivial interval
X
⊂
R
X \subset {\mathbf {R}}
and let
f
∈
C
1
(
X
,
X
)
f \in {C^1}(X,\,X)
be a chaotic mapping. We denote by
A
∞
(
f
)
{A_\infty }(f)
the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of
f
f
or to a subset of the absorbing boundary of
X
X
for
f
f
. A. We assume that
f
f
satisfies the following conditions: (1) the set of asymptotically stable periodic points for
f
f
is compact (an empty set is allowed), and (2)
A
∞
(
f
)
A{\,_\infty }(f)\,
is compact,
f
f
is expanding on
A
∞
(
f
)
{A_\infty }(f)
. Then we can associate a matrix
A
f
{A_f}
with entries either zero or one to the mapping
f
f
such that the number of periodic points for
f
f
with period
n
n
is equal to the trace of the matrix
[
A
f
]
n
{\left [ {{A_f}} \right ]^n}
; furthermore the zeta function of
f
f
is rational having the eigenvalues of
A
f
{A_f}
as poles. B. We assume that
f
∈
C
3
(
X
,
X
)
f \in {C^3}(X,\,X)
such that: (1) the Schwarzian derivative of
f
f
is negative, and (2) the closure of
A
∞
(
f
)
{A_\infty }(f)
is compact and
f
′
(
x
)
≠
0
f’ (x) \ne 0
for all
x
x
in the closure of
A
∞
(
f
)
{A_\infty }(f)
. Then we obtain the same result as in A.