We study the dynamics of towers defined by fixed points of renormalization for Feigenbaum polynomials in the complex plane with varying order $\ell $ of the critical point. It is known that the measure of the Julia set of the Feigenbaum polynomial is positive if and only if almost every point tends to $0$ under the dynamics of the tower for corresponding $\ell $ . That in turn depends on the sign of a quantity called the drift. We prove the existence and key properties of absolutely continuous invariant measures for tower dynamics as well as their convergence when $\ell $ tends to $\infty $ . We also prove the convergence of the drifts to a finite limit, which can be expressed purely in terms of the limiting tower, which corresponds to a Feigenbaum map with a flat critical point.

中文翻译：

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复杂费根鲍姆映射的极限漂移

我们研究了由变阶复平面中的 Feigenbaum 多项式的重整化不动点定义的塔的动力学$\ell $的临界点。众所周知，费根鲍姆多项式的 Julia 集的测度是正的当且仅当几乎每个点都趋于$0$在塔的动力学下为相应的$\ell $. 这又取决于称为漂移. 我们证明了塔动力学绝对连续不变测度的存在性和关键性质，以及它们的收敛性$\ell $倾向于$\infty $. 我们还证明了漂移收敛到一个有限极限，它可以纯粹用极限塔来表示，它对应于具有平坦临界点的费根鲍姆图。