We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

We prove that any Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.

The preperiodic dynatomic curve ${}_{n,p}$ is the closure in ℂ² of the set of (c,z) such that z is a preperiodic point of the polynomial $z\mapsto z^d+c$ with preperiod n and period p (n,p ≥ 1). We prove that each ${}_{n,p}$ has exactly d-1 irreducible components, which are all smooth and have pairwise transverse intersections at the singular points of ${}_{n,p}$. We also compute the genus of each component and the Galois group of the defining polynomial of ${}_{n,p}$.

We let $𝕃$ be the completion of the field of formal Puiseux series and study polynomials with coefficients in $𝕃$ as dynamical systems. We give a complete description of the dynamical and parameter space of cubic polynomials in $𝕃\left[\zeta \right]$. We show that cubic polynomial dynamics over $𝕃$ and $ℂ$ are intimately related. More precisely, we establish that some elements of $𝕃$ naturally correspond to the Fourier series of analytic almost periodic functions (in the sense of Bohr) which parametrize (near infinity) the quasiconformal...