源于簇代数突变的平面双有理映射的复动力学视角
Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations
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中文总结 AI 辅助
研究源自簇代数理论和可积系统的平面双有理映射，用全纯动力学方法计算其动力学次数，证明许多映射无守恒量和不变纤维化，回答相关问题，并应用遍历理论结果产生具有正熵和正李雅普诺夫指数的不变测度。
中文摘要 AI 辅助
我们使用全纯动力学方法研究源自簇代数理论和可积系统的平面双有理映射。通过计算这些映射的动力学次数（其中许多大于1），我们证明许多映射没有守恒量（也没有不变纤维化）。在大多数例子中，通过找到超吸引周期点也可以排除不变纤维化。这回答了马恰克和奥温豪斯（2024年）以及陈和李（2024年）提出的问题。此外，找到映射的良好代数稳定模型并计算动力学次数后，我们可以应用双有理映射遍历理论的结果来产生具有正熵和正李雅普诺夫指数的不变测度。
英文摘要
Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.
