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Chebotarov continua, Jenkins-Strebel differentials and related problems: a numerical approach

Marco Bertola

Abstract

We detail a numerical algorithm and related code to construct rational quadratic differentials on the Riemann sphere that satisfy the Boutroux condition. These differentials, in special cases, provide solutions of (generalized) Chebotarov problem as well as being instances of Jenkins--Strebel differentials. The algorithm allows to construct Boutroux differentials with prescribed polar part, thus being useful in the theory of weighted capacity and Random Matrices.

Chebotarov continua, Jenkins-Strebel differentials and related problems: a numerical approach

Abstract

We detail a numerical algorithm and related code to construct rational quadratic differentials on the Riemann sphere that satisfy the Boutroux condition. These differentials, in special cases, provide solutions of (generalized) Chebotarov problem as well as being instances of Jenkins--Strebel differentials. The algorithm allows to construct Boutroux differentials with prescribed polar part, thus being useful in the theory of weighted capacity and Random Matrices.
Paper Structure (17 sections, 2 theorems, 22 equations, 13 figures, 1 table)

This paper contains 17 sections, 2 theorems, 22 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Boutroux quadratic differentials.
  3. Main goal.
  4. Acknowledgements.
  5. Boutroux differentials with prescribed polar part at $\infty$
  6. General abstract considerations.
  7. Gradient descent analysis.
  8. Initialization.
  9. Main loop.
  10. Exit and other considerations.
  11. Comments on implementation
  12. Choice of branch-cuts and definition of the radical.
  13. Integration over $\mathscr R$.
  14. Special case: Chebotarov problems
  15. Pictures at an Exhibition
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any choices of sets $\mathcal{P}, \mathcal{E}$ and perimeters $a_j>0$ there is a unique Jenkins--Strebel differential as in Def. defJS such that, in addition, the complement of the critical graph $\Gamma$ is a union of $K$ disjoint disks $\mathbb D_j$ with $p_j\in\mathbb D_j$.

Figures (13)

  • Figure 1: In these two examples of the generalized Chebotarov problem where the set $\mathcal{E}$ is the same and we have two distinct solutions of the Boutroux problem \ref{['problemPhiE']} with $L=2$ stagnation points.
  • Figure 2: Chebotarov continuum for the set $\mathcal{E}=\{-1+1i,-1-1i,0.4+0.2i,2-1i,1+1i\}$ with $L=0$.
  • Figure 3: Chebotarov continuum for the set $\mathcal{E}=\{-1+1i,-1-1i,0.4+0.2i,2-1i,1+1i\}$ with $L=1$.
  • Figure 4: In this case $\Phi = \sum_{\ell=1}^1 t_\ell z^\ell$ with $[t_1,\dots,t_1]=[1]$. Here $t_0=0$. The set $\mathcal{E}=\{0-1i,0+1i,1+0i\}$ and $L=0$.
  • Figure 5: In this case $\Phi = \sum_{\ell=1}^3 t_\ell z^\ell$ with $[t_3,\dots,t_1]=[1+0i,0+0i,0-1i]$. Here $t_0=0$. The set $\mathcal{E}=\{0-1i,0+1i\}$ and $L=1$.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Definition 1.1: ArbarelloCornalba, Def 31
  • Theorem 1.1: ArbarelloCornalba Theroem 32
  • Theorem 1.2
  • Definition 1.2