Joint work with Michael Lyubich and Luna Lomonaco.

Douady and Hubbards notion of polynomial-like mappings $f : U \to U', f$ proper and $\newcommand{\D}{\mathbb{D}}U\simeq U'\simeq \D$ has proven to be very potent in the study of holomorphic dynamical systems. The main dynamical theorem is that any polynomial-like map of degree $d>1$ is hybridly equivalent to a degree $d$ polynomial, which is unique up to affine conjugacy, when the filled Julia set or the non-escaping set $K_f$ is connected. Here hybrid equivalence means conjugate by a q-c homeomorphism, which is conformal a.e. on $K_f$. This implies that a polynomial like map gives rise to two maps, an inner class the dynamics on the filled Julia set $K_f$, which is in the conformal class of some polynomia of the same degreel. And an outer or external class on the complement, which is in the conformal class of some same degree expanding circle map. The latter in turn is quasi-conformally conjugate to $z\mapsto z^d$. In other words a polynomial-like map is the restriction of a polynomial to some disk-like neighbourhood of its filled Julia set dressed up in some fancy "skirt".

A quadratic-like map is a degree $2$ polynomial-like mapping. The notion Mandelbrot-like family generalises the quadratic family $Q_c(z) = c + z^2$, $\newcommand{\C}{\mathbb{C}}c\in\C$ with connectedness locus the Mandelbrot set $M = \{ c\in \C | K_c \textrm{is connected}\}$. A Mandelbrot-like family is a holomorphic family of quadratic-like maps $f_\lambda : U_\lambda \to U'_\lambda$ parametrised by a disk $\Lambda\subset\C$ such that the connectedness locus $M_\Lambda$ is compactly contained in $\Lambda$ and homeomorphic to the Mandelbrot set. Douady and Hubbard proved that such a homeomorphism can always be chosen to preserve hybrid classes.

Lyubich proved that the hybrid class preserving homeomorphism between the connectedness loci of any two Mandelbrot-like families is quasi-conformal. In this talk I will discuss a theorem providing a bound on the local real dilatation of this parameter q-c homeomorphism in terms of the real dilatation of a hybrid conjugacy between the two corresponding polynomial-like maps.

This theorem can be viewed either as a corollary of or as a one-dimensional instance of a more general immersion theorem for Lyubich space QG of quadratic-like germs. In this talk I will focus on the second view point with a view towards the first.

The setting is Lyubich notion of quadratic-like germs. Lyubich constructed a natural equivalence on normalised quadratic-like mappings (i.e. of the form $\newcommand{\OO}{{\mathcal{O}}} z\mapsto c + z^2 + \OO(z^3)$ in $\C$ with equivalence classes denoted quadratic-like germs. He provided the space QG of quadratic-like germs with a complex analytic structure and showed that for this structure the space QG comes with two transversal holomorphic (partial) foliations. A horizontal foliation (only) of the connectedness locus with each leaf intersecting the Mandelbrot set in precisely one point and a globally defined vertical foliation by simply connected Riemann surfaces, each one corresponding to some fixed external class.

We can thus view the space QG as a book with very on-evenly sized and shaped vertical pages tied together by horizontal strands through the points of the Mandelbrot set. And a Mandelbro-like family gives after appropriate normalisation (affinely conjugating so that $f_\lambda(z) = c(\lambda) + z^2 + \OO(z^3)$) a mapping into QG, which is an immersion above the connectedness locus.

The general immersion theorem referred to above states that the space QG can be immersed into $H_0 \times \C$ by a fiberwise immersion respecting the vertical foliations, where $H_0$ is the zero horizontal leaf consisting of equivalence classes of maps with a super attracting fixed point.