On the regular leaf space of the cauliflower (Q1433497)

This article is devoted to the study of the regular leaf space of quadratic polynomials. The regular leaf space associated to a rational fraction \(f\) is a subset of the natural extension : it consists in the (negative) orbits \(\hat x\) such that all the inverse branches \(f^{-n}\) (along \(\hat x\)) are defined on an open set \(U\) independent of \(n\). Note that this space appears in the construction of the hyperbolic orbifold \(3\)-lamination associated to \(f\), due to \textit{M. Lyubich} and \textit{Y. Minsky} [J. Differ. Geom. 47, 17--94 (1997; Zbl 0910.58032)]. We note \(L_c\) the regular leaf space of \(P_c(z) = z^2 + c\). If \(c\) belongs to the main cardioid of the Mandelbrot set, \(L_c\) is homeomorphic to \(L_0\), which is the 2-dimensional extension of the 2-adic solenoid. This article is devoted to the description of the regular leaf space for \(c = 1/4\) (which is in the boundary of the Mandelbrot set). The method consists in exhibiting a pinching semiconjugacy from \(P_c\) (with \(c \in [0,1/4[\)) to \(P_{1/4}\). The author constructs dynamical tilings of the interior of the filled Julia set, and the semiconjugacy is obtained by gluing tile-to-tile homeomorphisms and the conjugacy between the dynamics of \(P_c\) and \(P_{1/4}\) outside the interior of the filled Julia set. One may deduce that the transversal structure of \(L_c\) does not change when \(c\) tends to 1/4 on the real axis, but the dynamics on the invariant leaves associated to the fixed points change from hyperbolic to parabolic.