Perfect subspaces of quadratic laminations

TL;DR

This paper explores quotient spaces derived from quadratic invariant laminations, providing new insights into the topology of the Mandelbrot set and related structures by 'unpinching' certain transitions.

Contribution

It introduces two new quotient spaces related to quadratic laminations, refining the understanding of the Mandelbrot set's combinatorial and topological structure.

Findings

The first quotient space simplifies the Mandelbrot set by unpinching hyperbolic component transitions.

The second quotient distinguishes renormalizable laminations, revealing finer topological distinctions.

Both quotients offer new perspectives on the structure of quadratic invariant laminations.

Abstract

The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to.