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#### Parabolic degrees and Lyapunov exponents for hypergeometric local systems

##### External Resource

https://doi.org/10.1080/10586458.2019.1580632

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Fougeron_Parabolic degrees and Lyapunov exponents for hypergeometric local systems_2021.pdf

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##### Citation

Fougeron, C. (2021). Parabolic degrees and Lyapunov exponents for hypergeometric local
systems.* Experimental Mathematics,* *30*(4), 531-546. doi:10.1080/10586458.2019.1580632.

Cite as: https://hdl.handle.net/21.11116/0000-0006-9C45-D

##### Abstract

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 - \alpha_i) - z \prod_{j=1}^h (\mathrm D - \beta_j) = 0$ where

$\mathrm D = z \frac {d}{dz}$. For $\alpha_i$ and $\beta_j$ 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.

(\mathrm D - \alpha_i) - z \prod_{j=1}^h (\mathrm D - \beta_j) = 0$ where

$\mathrm D = z \frac {d}{dz}$. For $\alpha_i$ and $\beta_j$ 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.