In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher dimensions.

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex Hénon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $(1+t)\lambda$, where $\lambda$ is a root of unity and $t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $|t|$, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and $J^{+}$ depend continuously on the parameters as $t\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on $J$ and $J^{+}$ when $t$ is nonnegative.

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials z¯d+c. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure, is the control of the position of these critical points.