Related papers: Compositional roots of H\'enon maps

Consider the standard family of complex H\'enon maps $H(x,y) = (p(x) - ay, x)$, where $p$ is a quadratic polynomial and $a$ is a complex parameter. Let $U^{+}$ be the set of points that escape to infinity under forward iterations. The…

As in [5], we study holomorphic maps of positive degree between compact complex manifolds, and prove that any holomorphic map of degree one from a compact complex manifold to itself is biholomorphic. This conclusion confirms that under a…

We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new…

We achieve compositions rules for the geometric parameters of the composed rotations, which is in a certain sense analogous to the well known Rodrigues formula. We also obtain a necessary and sufficient condition for a composition of two…

We consider links of complex isolated hypersurface singularities in $\mathbb{C}^{n+1}$ and study differentiable maps defined by restricting holomorphic functions to the links. We give an explicit example in which such a restriction gives a…

Given a reflection group $G$ acting on a complex vector space $V$, a reflection map is the composition of an embedding $X \hookrightarrow V$ with the orbit map $V\to\mathbb C^p$ that maps a $G$-orbit to a point. Reflection maps can be very…

We study Henon maps which are perturbations of a hyperbolic polynomial p with connected Julia set. We give a complete description of the critical locus of these maps. In particular, we show that for each critical point c of p, there is a…

In the paper, we first classify all polynomial maps $H$ of the following form: $H=\big(H_1(x_1,x_2,\ldots,x_n),H_2(x_1,x_2),H_3(x_1,x_2),\ldots,H_n(x_1,x_2)\big)$ with $JH$ nilpotent. After that, we generalize the structure of $H$ to…

Let $K$ be a field of characteristic 0 containing all roots of unity. We classify all the Hopf structures on monomial $K$-coalgebras, or, in dual version, on monomial $K$-algebras.

For area-preserving H\'enon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, the…

Let $F$ and $G$ be generalized H\'enon maps. We show that the Kato surfaces associated to the germs of $F$ and $G$ near infinity are biholomorphic if and only if $F$ and $G$ are conjugate in $\text{Aut}(\mathbb{C}^2)$. This answers a…

We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the…

For the finite ordered sets $A, D$, write $A^D$ for the ordered set of isotone maps $D \to A$ with the pointwise order. It was proved in earlier work that the order structure of $A^A$ determines~$A$ up to isomorphism. In this note we extend…

The polynomials with quaternion coefficients have two kind of roots: isolated and spherical. A spherical root generates a class of roots which contains only one complex number $z$ and its conjugate $\bar{z}$, and this class can be…

The objects of study in this paper are Hopf algebras $H$ which are finitely generated algebras over an algebraically closed field and are extensions of a commutative Hopf algebra by a finite dimensional Hopf algebra. Basic structural and…

We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).

Fold maps are fundamental tools in the theory of singularities of differentiable maps and its applications to geometry. They are higher dimensional variants of Morse functions. Classes of special generic maps and round fold maps are…

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can…

In this paper, we discuss the associated family of harmonic maps $\mathcal{F}: M \rightarrow G/K$ from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type which are either algebraic or totally symmetric. These…

We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…