John Erik Fornaess
In this paper we investigate the support of the unique measure of maximal entropy of complex Henon maps, J^*. The main question is whether this set is the same as the analogue of the Julia set, J.
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Rigidity of the Julia set for Hénon-Sibony maps
本文研究了Hénon-Sibony映射的Julia集刚性问题，证明若两个映射具有相同的正向Julia集，则它们共享一个公共迭代，扩展了Lamy在二维情况下的结果。
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 automorphism of positive entropy on a compact Kaehler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Henon type maps on C^k and of automorphisms of compact Kaehler manifolds.
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 $(1+t)λ$, where $λ$ 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 hyperb…
There are two natural definitions of the Julia set for complex Hénon maps: the sets $J$ and $J^\star$. Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set $J^\star$, under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in the paper [For06], but the proof is incomplete. Our proof closely follows ideas from [For06], deviating at two points where substantial dissipativity is used. We show that $J = J^\star$ al…
Consider the parameter space $\mathcal{P}_λ\subset \mathbb{C}^{2}$ of complex Hénon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $λ=e^{2πi p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_λ$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $λ$. We prove that there is an open disk of parameters in $\mathcal{P}_λ$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+…
The McMullen-Sullivan holomorphic motion for topologically conjugate, complex polynomials with connected Julia set follows level sets of the Bottcher coordinate. The Buzzard-Verma holomorphic motion for hyperbolic, unstably connected, polynomial diffeomorphisms of C^2 follows level sets of the Bedford-Smillie solenoid map. It follows that this solenoid map is a conjugacy for those Henon maps that are perturbations of (one-dimensional) hyperbolic maps with connected Julia set.
The purpose of this note is to explore further the rigidity properties of Hénon maps from arXiv:1806.08189. For instance, we show that if $H$ and $F$ are Hénon maps with the same Green measure ($μ_H=μ_F$), or the same filled Julia set ($K_H=K_F$), or the same Green function ($G_H=G_F$), then $H^2$ and $F^2$ have to commute. This, in turn, gives that $H$ and $F$ have the same non-escaping sets. Further we prove that, either of the association of a Hénon map $H$ to its Green measure $μ_H$ or to its filled Julia set $K_H$ or to its Green function $G_H$ is locally injective.
In this work we consider a class of endomorphisms of $\mathbb{R}^2$ defined by $f(x,y)=(xy+c,x)$, where $c\in\mathbb{R}$ is a real number and we prove that when $-1<c<0$, the forward filled Julia set of $f$ is the union of stable manifolds of fixed and $3-$periodic points of $f$. We also prove that the backward filled Julia set of $f$ is the union of unstable manifolds of the saddle fixed and $3-$periodic points of $f$.
Horn maps of semi-parabolic Hénon maps
本文证明了半抛物Hénon映射的Horn映射满足弱Ahlfors岛性质，从而统一了Julia集的两种定义，并证明小扰动可使前向Julia集的Hausdorff维数任意接近4。
Transcendental Hénon maps are the natural extensions of the well investigated complex polynomial Hénon maps to the much larger class of holomorphic automorphisms. We prove here that transcendental Hénon maps always have non-trivial dynamical behavior, namely that they always admit both periodic and escaping orbits, and that their Julia sets are non-empty and perfect.
Backward Julia sets for a class of p-adic Hénon like maps
本文研究了一类p进Hénon型映射的逆填充Julia集，证明了当参数c的绝对值小于等于1时，该集有界；而当绝对值大于1时，该集无界且具有无穷测度。
We prove a version of the classical $λ$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.
Computability of Julia sets for complex Hénon maps: The role of attracting and neutral cycles
该研究将复Hénon映射Julia集的可计算性从双曲情形扩展到含吸引盆地的非双曲情形，基于鞍点稳定流形的算法实现，同时指出中性动力学下高维存在部分类似一维的可计算性二分法。
We prove that the characterization of the critical locus for complex Hénon maps that are small perturbations of quadratic polynomials with disconnected Julia sets given by Firsova holds in a much larger HOV-like region from the complex horseshoe locus. The techniques of this paper are non-perturbative.
We consider the structure of substantially dissipative complex Hénon maps admitting a dominated splitting on the Julia set. The dominated splitting assumption corresponds to the one-dimensional assumption that there are no critical points on the Julia set. Indeed, we prove the corresponding description of the Fatou set, namely that it consists of only finitely many components, each either attracting or parabolic periodic. In particular there are no rotation domains, and no wandering components. Moreover, we show that $J = J^\star$ and the dynamics on $J$ is hyperbolic away from parabolic cycle…
We give a topological model of the critical locus for complex Hénon maps that are perturbations of the quadratic polynomial with disconnected Julia set.
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 share the same backward and forward Julia sets (or non-escaping sets). Secondly, we study the relationship between any pair of skew products of Hénon maps in $\mathbb{C}^3$ having the same forward and backward Julia sets.
We study the dynamics of the Hénon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Hénon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Hénon maps over ${\mathbb Q}_3$, this attractor is infinite and supports an SRB…
We show, for a class of automorphisms of C^k, that their equilibrium measures are exponentially mixing. In particular, this holds for (generalized) Henon maps.
Very little is currently known about the dynamics of non-polynomial entire maps in several complex variables. The family of transcendental Hénon maps offers the potential of combining ideas from transcendental dynamics in one variable, and the dynamics of polynomial Hénon maps in two. Here we show that these maps all have infinite topological and measure theoretic entropy. The proof also implies the existence of infinitely many periodic orbits of any order greater than two.
Julia sets and bifurcation loci
本文证明复平面和复二维空间中不同多项式动力系统产生的分形对象不可等同，特别指出 Misiurewicz PCF 三次多项式的强分歧轨迹闭包不能作为正则多项式自同态的 Julia 集，并证明 Hénon 映射与多项式自同态的 Julia 集不能重合。
We show that the dynamics of sufficiently dissipative semi-Siegel complex Hénon maps with golden-mean rotation number is not $J$-stable in a very strong sense. By the work of Dujardin and Lyubich, this implies that the Newhouse phenomenon occurs for a dense $G_δ$ set of parameters in this family. Another consequence is that the Julia sets of such maps are disconnected for a dense set of parameters.
Hénon maps: a list of open problems
本文汇集并提出 Hénon 映射在实、复、代数及算术动力学方面的开放问题，为后续研究提供方向。
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 occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering domains. In trying to find analogues of these one dimensional results, it is not clear which higher dimensional transcendental maps to consider. In this paper we find inspiration from the extensive work on the dynamics…
Mixing and CLT for Hénon-Sibony maps: plurisubharmonic observables
本文证明复Hénon映射及更一般的Hénon-Sibony映射的最大熵测度对多次调和可观测函数具有指数混合性和中心极限定理。
On the support of measures of large entropy for Hénon-Sibony maps
研究Hénon-Sibony映射的大熵不变测度的支撑集，证明若测度熵大于某阈值则支撑在Julia集上。
Multiplier rigidity for complex Hénon maps
复Hénon映射的乘子刚性问题，通过乘子谱或不稳定乘子确定，且在多次数和多Jacobian固定时保持刚性。
Computability properties of hyperbolic complex Hénon maps
本文建立了复Hénon映射（双曲多项式微分同胚）的Julia集可计算性、双曲性半可判定性以及参数空间中双曲性轨迹的下可计算性。
Quantitative Dynamics of complex Hénon maps
本文为复杂Hénon映射的动力学对象提供定量结果，作为具有几何控制的Pesin理论的定量版本，并证明周期点指数快速等分布至平衡测度。
We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.
Many results of the Fatou-Julia iteration theory of rational functions extend to uniformly quasiregular maps in higher dimensions. We obtain results of this type for certain classes of quasiregular maps which are not uniformly quasiregular.
In this paper we study the dynamics of regular polynomial automorphisms of C^n. These maps provide a natural generalization of complex Henon maps in C^2 to higher dimensions. For a given regular polynomial automorphism f we construct a filtration in C^n which has particular escape properties for the orbits of f. In the case when f is hyperbolic we obtain a complete description of its orbits. In the second part of the paper we study the Hausdorff and box dimension of the Julia sets of f. We show that the Julia set J has positive box dimension, and (provided f is not volume preserving) that the …
Consider the standard family of complex Hénon 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 analytic structure of the escaping set $U^{+}$ is well understood from previous work of J. Hubbard and R. Oberste-Vorth as a quotient of $(\mathbb{C}-\overline{\mathbb{D}}) \times\mathbb{C}$ by a discrete group of automorphisms $Γ$ isomorphic to $\mathbb{Z}[1/2]/\mathbb{Z}$. On the other hand, the boundary $J^{+}$ of $U^{+}$ is a complicated fractal object on which…
A thermodynamic path metric for complex Hénon maps
本文为复杂Hénon映射参数空间中的双曲分量构造了一个Hermite协方差形式，并证明其诱导距离，给出了压力型度量结构的高维全纯动力对应。
We consider the family of Henon maps in the plane and show that the SRB measures vary continuously in the weak* topology within the set of Benedicks-Carleson parameters.
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 primary component of the critical locus asymptotic to the line y = c. Moreover, primary components are conformally equivalent to the punctured disk, and their orbits cover the whole critical set. We also describe the holonomy maps from such a component to itself along the leaves of two natural foliations. Finally, we show that a quadratic Henon map taken along…
We give three families of parabolic rational maps and show that every Cantor set of circles as the Julia set of a non-hyperbolic rational map must be quasisymmetrically equivalent to the Julia set of one map in these families for suitable parameters. Combining a result obtained before, we give a complete classification of the Cantor circles Julia sets in the sense of quasisymmetric equivalence. Moreover, we study the regularity of the components of the Cantor circles Julia sets and establish a sufficient and necessary condition when a component of a Cantor circles Julia set is a quasicircle.
For complex parameters a,c, we consider the Henon mapping H_{a,c}: C^2 -> C^2 given by (x,y) -> (x^2 +c -ay, x), and its Julia set, J. In this paper, we describe a rigorous computer program for attempting to construct a cone field in the tangent bundle over J, which is preserved by DH, and a continuous norm in which DH (and DH^{-1}) uniformly expands the cones (and their complements). We show a consequence of a successful construction is a proof that H is hyperbolic on J. We give several new examples of hyperbolic maps, produced with our computer program, Hypatia, which implements our methods.
We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable Hénon-like maps, parametrised by a quantity called the `average Jacobian' $b$, the set of all parameters $b$ such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.
We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the Hénon and the Lozi maps.
It was known that the Sierpiński carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well known rational maps, such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpiński carpets.
For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which…
In studying the complex Hénon maps, Mummert (in "Holomorphic shadowing for Hénon maps" Nonlinearity 21 pp. 2887-2898, 2008) defined an operator the fixed points of which give rise to bounded orbits. This enabled him to obtain an estimate of the solenoid locus. Instead of the contraction mapping theorem, in this paper we present an implicit function theorem version of his result by providing an alternative proof of a theorem of Hubbard and Oberste-Vorth (in Real and Complex Dynamical Systems, pp.89-132, 1995).
Rigidity of the escaping set of certain Hénon maps
该研究证明了特定Hénon映射的逃逸集在复平面自同构作用下的刚性，并展示了其自同构群的结构。
Local connectivity of Julia sets of some rational maps with Siegel disks
本文证明有理映射长迭代在有界型Siegel盘边界附近扩张，借此将Petersen关于二次Siegel多项式Julia集局部连通性的结果推广至一般情形，且证明无需使用puzzles。
Sun Daochun and Yang Lo have shown that many results of the Fatou-Julia iteration theory of rational functions extend to quasiregular self-maps of the Riemann sphere for which the degree exceeds the dilatation. We show that in this context, in contrast to the case of rational functions, the Julia set may have Hausdorff dimension zero. On the other hand, we exhibit a gauge function depending on the degree and the dilatation such that the Hausdorff measure with respect to this gauge function is always positive, but may be finite.
We extend the renormalisation operator introduced in \cite{dCML} from period-doubling Hénon-like maps to Hénon-like maps with arbitrary stationary combinatorics. We show the renormalisation picture holds also holds in this case if the maps are taken to be \emph{strongly dissipative}. We study infinitely renormalisable maps $F$ and show they have an invariant Cantor set $\mathcal{O}$ on which $F$ acts like a $p$-adic adding machine for some $p>1$. We then show, as for the period-doubling case in \cite{dCML}, the sequence of renormalisations have a universal form, but the invariant Cantor set $\…
Marker endomorphisms for complex Hénon maps
本文研究复Hénon映射中复合标记自同态的性质，提出检验单个标记自同态双射性的可计算准则，并利用Hedlund的置换函数重新表述以应用其结论。
The Julia set of the exponential family $E_κ:z\mapstoκe^z$, $κ>0$ was shown to be the entire complex plane when $κ>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<κ\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points…
Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We show that under suitable assumptions the Julia set of $f$ has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $f^n(z)$ converges to zeros of $g$ almost everywhere in $\mathbb{C}$ if this is the case for each zero of $g''$. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.
We study the quasisymmetric geometry of the Julia sets of McMullen maps $f_λ(z)=z^m+λ/z^\ell$, where $\ell$, $m\geq 2$ are integers satisfying $1/\ell+1/m<1$ and $λ\in\mathbb{C}\setminus\{0\}$. If the free critical points of $f_λ$ are escaped to the infinity, we prove that the Julia set $J_λ$ of $f_λ$ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a sufficient condition on $λ$ such that $J_λ$ is a Sierpiński carpet and pro…
Julia sets and geometrically finite maps over finite extensions of the $p$-adic field
本文研究p-进数域有限扩张上有理映射的Julia集，证明在临界轨道良好时K-Julia集为C_p-Julia集的自然限制，并在几何有限条件下证明其动力学为可数状态马尔可夫移位。
In this paper we analyze chaotic dynamics for two dimensional nonautonomous maps through the use of a nonautonomous version of the Conley-Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley-Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the…
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties…
In this paper we study the locus of generalized degree $d$ Henon maps in the parameter space $\operatorname{Rat}_d^N$ of degree $d$ rational maps $\mathbb{P}^N\to\mathbb{P}^N$ modulo the conjugation action of $\operatorname{SL}_{N+1}$. We show that Henon maps are in the GIT unstable locus if $N\ge3$ or $d\ge3$, and that they are semistable, but not stable, in the remaining case of $N=d=2$. We also give a general classification of all unstable maps in $\operatorname{Rat}_2^2$.
Uniformization of Julia sets of geometrically finite rational maps
该研究刻画了Julia集与Schottky集拟共形等价的几何有限有理映射，证明了几何有限单临界多项式无穷远吸引域闭包与Schottky集拟共形等价的充要条件，并给出了拟圆周对的相关动力学刻画。
The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.
We effect a multifractal analysis for a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We decompose the set of non wandering points on the unstable manifold into level sets of Birkhoff averages of continuous functions, and derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures.
We consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of qua…
The pruning front conjecture, folding patterns and classification of Hénon maps in the presence of strange attractors
本研究针对Wang-Young参数集下的Hénon映射，证明了修剪前沿猜想、建立揉理论，给出基于折叠模式与揉序列的奇异吸引子共轭分类，结果基于密分支树的逆极限描述拓展得到。
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