On the dynamical degrees of meromorphic maps preserving a fibration

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Let f be a dominant meromorphic self-map on a projective manifold X which preserves a meromorphic fibration pi: X --> Y of X over a projective manifold Y. We establish formulas relating the dynamical degrees of f, the dynamical degrees of f relative to the fibration and the dynamical degrees of the self-map g on Y induced by f. Applications are given.

Lie Symmetries of Birational Maps Preserving Genus 0 Fibrations

We prove that any planar birational integrable map, which preserves a fibration given by genus $0$ curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods of the considered maps.

Algebraic stability of meromorphic maps descended from Thurston's pullback maps

Let $ϕ:S^2 \to S^2$ be an orientation-preserving branched covering whose post-critical set has finite cardinality $n$. If $ϕ$ has a fully ramified periodic point $p_{\infty}$ and satisfies certain additional conditions, then, by work of Koch, $ϕ$ induces a meromorphic self-map $R_ϕ$ on the moduli space $\mathcal{M}_{0,n}$; $R_ϕ$ descends from Thurston's pullback map on Teichmüller space. Here, we relate the dynamics of $R_ϕ$ on $\mathcal{M}_{0,n}$ to the dynamics of $ϕ$ on $S^2$. Let $\ell$ be the length of the periodic cycle in which the fully ramified point $p_{\infty}$ lies; we show that $R…

超 Kähler 流形的抛物自同构

Parabolic automorphisms of hyperkahler manifolds

本文研究超 Kähler 流形上的抛物自同构，证明保持 Lagrangian 纤维化者在纤维上遍历作用，并将 K3 曲面的相关结果推广至一般超 Kähler 情形。

具有有理不变纤维化的Kahan-Hirota-Kimura映射的动力学

Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations

本文提出一种研究平面Kahan-Hirota-Kimura映射动力学的方法，展示可积KHK映射可能表现出复杂动力学，即使其来源于行为简单的向量场。通过研究二次平面向量场的KHK映射，证明其保持原第一积分并具有向量场作为李对称性，且在密集的积分步长下全局周期性。

保体积流的缠绕性（Wrappingness）与主干性（Trunkenness）

The Wrappingness and Trunkenness of Volume-Preserving Flows

本文推广链环不变量概念，定义3维流形上保体积流的微分同胚不变量缠绕性与主干性，证明其为流的全局截面曲面存在的阻碍，且二者并非流的螺旋度的函数。