The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$ , with the property that there are infinitely many hypersurfaces on  $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$ . In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$ -varieties and of Cantat’s theorem to self-correspondences.

下面的定理，其中包括作为非常特殊情况的 Jouanolou 和 Hrushovski 的结果，一方面是代数$D$ -varieties，另一方面是 Cantat 在理性动力学上的结果： 在特征为零的域上工作，假设$ \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$是从有限类型（可能是非约简）不可约方案$Z$到一个代数变体$X$ ，其性质是在$X$上有无限多个超曲面， 其方案理论逆图像在$\unicode[STIX]{x1D719}_{1}$和$\unicode[STIX]{x1D719 }_{2}$ 同意。然后在$X$上有一个非常数有理函数$g$使得$g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$。在$Z$也减少的情况下，可以用适当的变换代替方案理论的逆图像。在积极的特性中获得了部分结果。应用包括将 Jouanolou–Hrushovski 定理扩展到广义代数${\mathcal{D}}$ -varieties 和将 Cantat 定理扩展到自对应。