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In this note, we discuss the possible existence of finite criticaltrajectories connecting two zeros a(t) and b(t) of a family of quadraticdifferentials satisfying some properties. We treat the cases of holomorphic andmeromorphic quadratic differentials, and we give new proofs concerning thesupports of limit measures of the root-counting measures of the generalizedLaguerre and Jacobi polynomials with varying parameters.

On the existence of finite critical trajectories in a family of quadratic differentials