  A tournament is \emph{acyclically indecomposable} if no acyclic autonomous
set of vertices has more than one element. We identify twelve infinite
acyclically indecomposable tournaments and prove that every infinite
acyclically indecomposable tournament contains a subtournament isomorphic to
one of these tournaments. The {\it profile} of a tournament $T$ is the function
$\phi_T$ which counts for each integer $n$ the number $\phi_T(n)$ of
tournaments induced by $T$ on the $n$-element subsets of $T$, isomorphic
tournaments being identified. As a corollary of the result above we deduce that
the growth of $\phi_T$ is either polynomial, in which case $\phi_T(n)\simeq
an^k$, for some positive real $a$, some non-negative integer $k$, or as fast as
some exponential.
