### Abstract

Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form (Formula Presented). Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.

Original language | English |
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Title of host publication | Trends in Mathematics |

Publisher | Springer International Publishing |

Pages | 369-438 |

Number of pages | 70 |

Edition | 9783319524696 |

DOIs | |

State | Published - Jan 1 2017 |

### Publication series

Name | Trends in Mathematics |
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Number | 9783319524696 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

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### Keywords

- Asymptotic root-counting measure
- Critical trajectories
- Jacobi polynomials
- Quadratic differentials

### Cite this

*Trends in Mathematics*(9783319524696 ed., pp. 369-438). (Trends in Mathematics; No. 9783319524696). Springer International Publishing. https://doi.org/10.1007/978-3-319-52471-9_23