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.

翻译：带有美元元元元元的有限动态系统是一个函数 $f: X\ to X1\ to X_x美元, 美元=X_ 1\times\ dotstime X_n美元, 美元=xx=X_ 1\ dotstimes X_n美元, 美元是整数的有限间隔产物。 这种系统的结构由签名的简记 $G$(称为互动图 ) 表示 美元: 美元是一个螺旋, 每部分一个, 签署的弧表示它们之间的正和负影响。 精度动态系统是基因网络的通常模式。 在这种情况下, 通常假设美元是 美元 美元 的平面互动 美元, 也就是说, 美元+1 美元 的大小最多为美元 。 假设美元是连接, 美元是受度限制的, 我们证明: 如果 美元不是周期, 那么美元+fn\\\\ x1} 可能是不变的。 那么, 美元是一种非常简单的动态 : 一种全球接近 美元 美元 美元 的固定点是 美元 。