AI 中文总结
本文在相对代数动力学框架下拓展了动力学次数理论，证明半共轭的相对动力学次数对应一般纤维上诱导的扭曲有理映射的动力学次数，还证得仿射簇扭曲自同态第一动力学次数的代数性结果。
AI 中文摘要
扭曲有理映射自然出现在相对代数动力学中：若一个有理自映射保持一个纤维化，则其在一般纤维上诱导的映射通常不是函数域上的普通有理自映射，而是一个扭曲有理映射。这表明扭曲有理映射构成了研究相对动力学的自然框架。在该框架下，我们拓展了动力学次数理论——动力学次数是衡量动力系统渐近复杂性的数值不变量：其定义极限存在、与极化的选择无关，且是双有理不变量。我们还将半共轭的相对动力学次数与一般纤维上诱导的扭曲有理映射的动力学次数对应起来，并证明了相应的混合次数公式。最后，利用动力学次数的谱解释和无穷远处的赋值方法，我们证明了仿射簇的扭曲自同态的第一动力学次数的代数性结果。
英文摘要
Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordinary rational self-map over the function field, but a twisted one. This suggests that twisted rational maps form a natural framework for studying relative dynamics. In this framework we extend the theory of dynamical degrees, the numerical invariants measuring the asymptotic complexity of a dynamical system: the defining limits exist, are independent of the choice of polarization, and are birational invariants. We also identify the relative dynamical degrees of a semi-conjugacy with the dynamical degrees of the induced twisted rational map on the generic fiber, and prove the corresponding mixed degree formula. Finally, using the spectral interpretation of dynamical degrees and valuative methods at infinity, we prove an algebraicity result for the first dynamical degree of twisted endomorphisms of affine varieties.
