We address the inverse problem for holomorphic germs of a mapping of the complex line near a fixed point which is tangent to the identity. We provide a preferred parabolic map $\Delta $ realizing a given Birkhoff–Écalle–Voronin modulus $\psi $ and prove its uniqueness in the functional class we introduce. The germ is the time-$1$ map of a Gevrey formal vector field admitting meromorphic sums on a pair of infinite sectors covering the Riemann sphere. For that reason, the analytic continuation of $\Delta $ is a multivalued map admitting finitely many branch points with finite monodromy. In particular, $\Delta $ is holomorphic and injective on an open slit sphere containing $0$ (the initial fixed point) and $\infty $, where the companion parabolic point is situated under the involution ${-1}/{\mathrm {Id}}$. One finds that the Birkhoff–Écalle–Voronin modulus of the parabolic germ at $\infty $ is the inverse $\psi ^{\circ -1}$ of that at $0$.

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$.

In this paper, we study the dynamics of the Newton maps for arbitrary polynomials. Let p be an arbitrary polynomial with at least three distinct roots, and f be its Newton map. It is shown that the boundary $\partial B$ of any immediate root basin B of f is locally connected. Moreover, $\partial B$ is a Jordan curve if and only if $\mathrm {deg}(f|_B)=2$. This implies that the boundaries of all components of root basins, for the Newton maps for all polynomials, from the viewpoint of topology, are tame.

We extend Thurston’s topological characterisation theorem for postcritically finite rational maps to a class of rational maps which have a fixed bounded type Siegel disk. This makes a small step towards generalizing Thurston’s theorem to geometrically infinite rational maps.

We show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

Let $f$ be an $n$ -dimensional holomorphic map defined in a neighborhood of the origin such that the origin is an isolated fixed point of all of its iterates, and let ${\mathcal{N}}_{M}(f)$ denote the number of periodic orbits of $f$ of period $M$ hidden at the origin. Gorbovickis gives an efficient way of computing ${\mathcal{N}}_{M}(f)$ for a large class of holomorphic maps. Inspired by Gorbovickis’ work, we establish a similar method for computing ${\mathcal{N}}_{M}(f)$ for a much larger class of holomorphic germs, in particular, having arbitrary Jordan matrices as their linear parts. Moreover, we also give another proof of the result of Gorbovickis [On multi-dimensional Fatou bifurcation. Bull. Sci. Math.138(3)(2014) 356–375] using our method.

We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk. We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holds in the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.

Let d(c) denote the Hausdorff dimension of the Julia set Jc of the polynomial fc (z) = z 2 +c. The function c ↦ d(c) is real-analytic on the interval (–3/4, 1/4), which is in the domain bounded by the main cardioid of the Mandelbrot set. We prove that the function d is convex close to 1/4 on the left side of it.

It is known that the Fatou set of the map exp(z)/z defined on the punctured plane ℂ* is empty. We consider the M-set of λ exp(z)/z consisting of all parameters λ for which the Fatou set of λexp(z)/z is empty. We prove that the M-set of λexp(z)/z has infinite area. In particular, the Hausdorff dimension of the M-set is 2. We also discuss the area of complement of the M-set.

We study the postcritically finite maps within the moduli space of complex polynomial dynamical systems. We characterize rational curves in the moduli space containing an infinite number of postcritically finite maps, in terms of critical orbit relations, in two settings: (1) rational curves that are polynomially parameterized; and (2) cubic polynomials defined by a given fixed point multiplier. We offer a conjecture on the general form of algebraic subvarieties in the moduli space of rational maps on ${ \mathbb{P} }^{1} $ containing a Zariski-dense subset of postcritically finite maps.

Let F(z) be a rational map with degree at least three. Suppose that there exists an annulus such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.