相关论文: Rigidity of the escaping set of certain H\'enon ma…

Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic…

For a H\'{e}non map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the…

For a H\'enon map of the form $H(x, y) = (y, p(y) - ax)$, where $p$ is a polynomial of degree at least two and $a \not= 0$, it is known that the sub-level sets of the Green's function $G^+_H$ associated with $H$ are Short $\mathbb C^2$'s.…

We prove that the Julia set of a Henon type automorphism on C^2 is very rigid: it supports a unique positive ddc-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an…

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…

The purpose of this note is to explore further the rigidity properties of H\'{e}non maps from arXiv:1806.08189. For instance, we show that if $H$ and $F$ are H\'{e}non maps with the same Green measure ($\mu_H=\mu_F$), or the same filled…

We investigate the multiplier rigidity problem for polynomial automorphisms of $\mathbf{C}^2$. A first result states that a complex H\'enon map of given degree is determined up to finitely many choices by its multiplier spectrum, or more…

The purpose of this note is two fold. First, we study the relation between a pair of H\'{e}non maps that share the same forward and backward non-escaping sets. Second, it is shown that there exists a continuum of $Short-\mathbb{C}^2$'s that…

Let $H$ and $F$ be two H\'{e}non maps with biholomorphically equivalent escaping sets, then there exist affine automorphisms $A_1$ and $A_2$ in $\mathbb{C}^2$ such that \[ F=A_1\circ H \circ A_2 \] in $\mathbb{C}^2$.

We extend the results of arXiv:2206.08295v2 by showing that any homothety in $\mathbb T^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small…

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that…

Transcendental H\'enon maps are the natural extensions of the well investigated complex polynomial H\'enon maps to the much larger class of holomorphic automorphisms. We prove here that transcendental H\'enon maps always have non-trivial…

We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also…

The goal of this article is two fold. Firstly, we explore the dynamics of a semigroup of polynomial automorphisms of $\mathbb{C}^2$, generated by a finite collection of H\'enon maps. In particular, we construct the positive and negative…

We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental H\'enon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general…

We show that every sufficiently large r-regular digraph G which has linear degree and is a robust outexpander has an approximate decomposition into edge-disjoint Hamilton cycles, i.e. G contains a set of r-o(r) edge-disjoint Hamilton…

We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature. Our main result applies to its possible homomorphisms into other…

We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of the complex H\'enon 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 H\'enon maps which…

In this paper we prove that the homotopy class of non-homothety linear endomorphisms on $\mathbb{T}^2$ with determinant greater than 2 contains a $C^1$ open set of non-uniformly hyperbolic endomorphisms. Furthermore, we prove that the…

Let G be a finite simple graph with automorphism group A(G). Then a spanning subgraph U of G is a fixing subgraph of G if G contains exactly $| A(G)|/ | A(G) \cap A(U)| $ subgraphs isomorphic to U: the graph G must always contain at least…