  A finite dynamical system with $n$ components is a function $f:X\to X$ where
$X=X_1\times\dots\times X_n$ is a product of $n$ finite intervals of integers.
The structure of such a system $f$ is represented by a signed digraph $G$,
called interaction graph: there are $n$ vertices, one per component, and the
signed arcs describe the positive and negative influences between them. Finite
dynamical systems are usual models for gene networks. In this context, it is
often assumed that $f$ is {\em degree-bounded}, that is, the size of each $X_i$
is at most the out-degree of $i$ in $G$ plus one. Assuming that $G$ is
connected and that $f$ is degree-bounded, we prove the following: if $G$ is not
a cycle, then $f^{n+1}$ may be a constant. In that case, $f$ describes a very
simple dynamics: a global convergence toward a unique fixed point in $n+1$
iterations. This shows that, in the degree-bounded case, the fact that $f$
describes a complex dynamics {\em cannot} be deduced from its interaction
graph. We then widely generalize the above result, obtaining, as immediate
consequences, other limits on what can be deduced from the interaction graph
only, as the following weak converses of Thomas' rules: if $G$ is connected and
has a positive (negative) cycle, then $f$ may have two (no) fixed points.
