超几何局部系统的抛物度和李雅普诺夫指数
Parabolic degrees and Lyapunov exponents for hypergeometric local systems
摘要：
考虑$\mathrm{CP}^1 - \{0,1,\infty \}$上的平坦丛对应于超几何微分方程$ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$的解，其中$\mathrm D = z \frac {d}{dz}$。对于$α_i$和$β_j$不同的实数，这个丛是已知的Hodge结构的复杂极化变化的基础。在$\mathrm{CP}^1 - \{0,1,\infty \}$上设置完整的双曲度量，我们将$n$李雅普诺夫指数与这个丛相关联。通过数值模拟，计算了由Hodge结构变化引起的全纯子丛的抛物度，并研究了李雅普诺夫指数与抛物度的关系.
英文摘要：
Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $α_i$ and $β_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations.
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