In this paper we will develop a general approach which shows that generalized "critical relations" of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide class of one-parameter families of interval and circle maps, for example for maps with flat critical points, but also for maps with complex analytic extensions such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1}, for real maps we obtain {\em positive} transversality (where $>0$ holds instead of just $\ne 0$), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family. This method additionally gives results for unimodal families of the form $x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$ real.

中文翻译：

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通过传递算子和全纯运动实现正横向性，并应用于区间图的单调性

在本文中，我们将开发一种通用方法，该方法表明复平面上局部定义的全纯映射族的广义“临界关系”横向展开。主要思想是定义一个传递算子，它是 Thurston 回拉算子的局部模拟，使用全纯运动。假设满足所谓的提升特性，我们将获得有关此传输算子的频谱的信息，从而获得有关横向的信息。我们方法的一个重要新特征是它不是全局的：我们考虑的地图只需要在某个有限集的邻域上定义和全纯。我们将通过获取区间和圆形地图的一大类单参数族的横向性来说明这种方法，例如对于具有平坦临界点的地图，但也适用于具有复杂解析扩展的地图，例如某些类似多项式的地图。正如在辻井的方法 \cite{Tsu0,Tsu1} 中一样，对于真实地图，我们获得了 {\em positive} 横向（其中 $>0$ 成立，而不仅仅是 $\ne 0$），因此这些家族的熵是单调性的，并且也（作为一个简单的应用程序）用于真正的二次族。该方法另外给出了 $x\mapsto |x|^\ell+c$ 形式的单峰族的结果，因为 $\ell>1$ 不一定是偶数整数和 $c$ 实数。