Collet, Eckmann and the bifurcation measure

The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $\mu_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $\mu_\mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $\mu_\mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps

On the dynamical Teichmüller space

We prove that the dynamical Teichmüller space of a rational map immerses into the space of rational maps of the same degree, answering a question of McMullen and Sullivan. This is achieved through a new description of the tangent and cotangent space to the dynamical Teichmüller space

Hyperbolicity and Bifurcations in holomorphic families of polynomial skew products

We study holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\Bbb{C}^2$ of the form $F(z,w)=(p(z),q(z,w))$ that extend to holomorphic endomorphisms of $\Bbb{P}^2(\Bbb{C})$. We prove that stability in the sense of [Berteloot, Bianchi, and Dupont, 2018] preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current near the boundary of the parameter space. One of our tools is an asymptotic equidistribution property for the bifurcation current. This is established in the general setting of families of endomorphisms of $\Bbb{P}^k$, and is the first equidistribution result of this kind for holomorphic dynamical systems in dimension larger than one

Bifurcation loci of families of finite type meromorphic maps

We show that $J-$ stability is open and dense in natural families of meromorphic maps of one complex variable with a finite number of singular values, and even more generally, to finite type maps. This extends the results of Ma\~{n}\'e-Sad-Sullivan for rational maps of the Riemann sphere and those of Eremenko and Lyubich for entire maps of finite type of the complex plane, and essentially closes the problem of density of structural stability for holomorphic dynamical systems in one complex variable with finitely many singular values. This result is obtained as a consequence of a detailed study of a new type of bifurcation that arises with the presence of both poles and essential singularities (namely periodic orbits exiting the domain of definition of the map along a parameter curve), and in particular its relation with the bifurcations in the dynamics of singular values. The presence of these new bifurcation parameters require essentially different methods to those used in previous work for rational or entire maps