Perfect subspaces of quadratic laminations

Abstract

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary is defined as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady (1984) and, separately, by Thurston (1985) who used quadratic invariant geolaminations as a major tool. We showed earlier that the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant geolaminations with the Hausdorff distance topology. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that used for the Mandelbrot set. The resulting quotient space is obtained from the Mandelbrot set by ipinching the transitions between adjacent hyperbolic components. In the second case we identify renormalizable geolaminations that can be irenormalized to the same hyperbolic geolamination while no two non-renormalizable geolaminations are identified.