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

The morphology of infinite tournaments. Application to the growth of their profile