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Regular version of the site

Working paper

Smart criticality

Blokh A., Oversteegen L., Ptacek R., Vladlen Timorin.
A crucial fact established by Thurston in his 1985 preprint is that distinct \emph{minors} of quadratic laminations do not cross inside the unit disk; this led to his construction of a combinatorial model of the Mandelbrot set. Thurston's argument is based upon the fact that \emph{majors} of a quadratic lamination never enter the region between them, a result that fails in the cubic case. In this paper, devoted to laminations of any degree, we use an alternative approach in which the fate of sets of intersecting leaves of two distinct laminations is studied. It turns out that, under some natural assumptions, these sets of intersecting leaves behave like gaps of a lamination. Relying upon this, we rule out certain types of mutual location of critical sets of distinct laminations (this can be viewed as a partial generalization of the theorem that quadratic minors do not cross inside the unit disk). The main application is to the cubic case.