Marco Martens, Liviana Palmisano, Zhuang Tao
It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of $\mathbb{C}^2$, with bounded degree, there are Newhouse laminations.
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Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor, non trivial analytic families of polynomial maps with infinitely many sinks.
Let $\mathcal{L}$ be a Lipschitz lamination by Riemann surfaces embedded in $M$. If $M$ is a complex torus, $\mathbb{CP}^1\times\mathbb{CP}^1$ or $\mathbb{T}^1\times\mathbb{CP}^1$ and there is no directed closed current then there exists a unique directed harmonic current of mass one. Moreover if $\mathcal{L}$ is embedded in $M=\mathbb{CP}^1\times\mathbb{CP}^1$ and has no compact leaves, then there is no directed closed current. If $\mathcal{L}$ is not Lipschitz, then slightly weaker results are obtained.
In this article, we show the existence of a nontrivial Riemann surface lamination embedded in $\mathbb{CP}^2$ by using Donaldson's construction of asymptotically holomorphic submanifolds. Further, the lamination we obtain has the property that each leaf is a totally geodesic submanifold of $\mathbb{CP}^2 $ with respect to the Fubini-Study metric. This may constitute a step in understanding the conjecture on the existence of minimal exceptional sets in $\mathbb{CP}^2$.
Inner Functions and Laminations
本文利用黎曼曲面层叠结构上的测地流和环面流研究内函数的轨道计数问题，证明了有限李雅普诺夫指数内函数的原像计数渐近公式，并揭示几乎每条逆轨道的迭代具有渐近线性性质。
To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$σ_d$-invariant laminations}, where $σ_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials $f(z) = z^2 +c$. In the spirit of Thurston's work, we consider the space of all \emph{cubic symmetric polynomials} $f_λ(z)=z^3+λ^2 z$ in a series of three articles. In the present paper, the first in the series, we construct a lamination $C_sCL$ together with the induced factor space ${\mathbb{S}}/C_sCL$ of the unit circle ${\mathbb{S}}$. As will be verified in the th…
Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps
本文揭示了紧双曲流形的$\mathbb{Z}$-覆盖中距离极小叶状结构与球面轨道闭包的联系，构造了具有指定性质的$\mathbb{Z}$-覆盖并明确描述了所有horocycle轨道闭包，同时证明双曲度量的微小扰动会导致其拓扑发生剧烈变化。
Limit cubic laminations
针对单位圆三倍映射的σ₃-不变树状层状结构，在给定极限临界肖像的前提下，研究其极限临界肖像，得到端点非周期时的唯一有限临界集层状结构及极端情形的可数闭临界肖像族，结果可用于构造三次连通轨迹模型。
Let $K \subseteq \mathbb{R}^{2 \times 2}$ be a compact set, let $K^{rc}$ be its rank-one convex hull, and let $L(K)$ be its lamination convex hull. It is shown that the mapping $K \to \overline{L(K)}$ is not upper semicontinuous on the diagonal matrices in $\mathbb{R}^{2 \times 2}$, which was a problem left by Kolář. This is followed by an example of a 5-point set of $2 \times 2$ symmetric matrices with non-compact lamination hull. Finally, an example of another 5-point set $K$ is given, which has $L(K)$ connected, compact and strictly smaller than $K^{rc}$.
Affine laminations and coaffine representations
该研究探讨SL(4,R)的曲面子群在RP³上的凸余紧作用，分析其弯曲数据空间ML^ρ(S)的射影化性质，得出其为6g-7维球面的结论。
In the space of polynomial maps of $\mathbb R^2$ of degree at least two, there are codimension $3$ laminations of maps with at least $3$ period doubling Cantor attractors. The leafs of the laminations are real-analytic and they have uniform diameter. The closure of each lamination contains the codimension one tangency locus of a saddle point. Asymptotically, the leafs of each lamination align with the leafs of the eigenvalue foliation. This is an example of general coexistence theorems valid for higher dimensional real-analytic unfoldings of two dimensional homoclinic tangencies.
We consider the problem of classifying the dynamics of complex polynomials $f: \mathbb{C} \to \mathbb{C}$ restricted to their basins of infinity. We synthesize existing combinatorial tools --- tableaux, trees, and laminations --- into a new invariant of basin dynamics we call the pictograph. For polynomials with all critical points escaping to infinity, we obtain a complete description of the set of topological conjugacy classes. We give an algorithm for constructing abstract pictographs, and we provide an inductive algorithm for counting topological conjugacy classes with a given pictograph.
Laminations and External Angles for Similarity Pairs
本研究针对复平面上由两个相似映射生成的相似对动力系统，证明边界Mandelbrot集参数下吸引子边界的半群作用拓扑共轭于分段线性作用，给出填充集含割点的充要条件并推广了二次次要层状结构理论。
Lavaurs algorithm for cubic symmetric polynomials
本文为三次对称多项式空间构造层叠$C_sCL$，提出类似Lavaurs算法的构造方法，为三次对称连通轨迹提供单调模型。
We show that there exists a polynomial automorphism $f$ of $\mathbb{C}^{3}$ of degree 2 such that for every automorphism $g$ sufficiently close to $f$, $g$ admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each $d \ge 2$, there exists an open set of polynomial automorphisms of degree at most $d$ in which the automorphisms having infinitely many sinks are dense. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz.
In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: let $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \[ f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) \longmapsto \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r) \\ r+\varepsilon v_{\pm 1}(θ,r) \end{array}\right), \] where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions \[ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0), \] where $ω_k =\pm 1$…
In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \begin{eqnarray} \nonumber f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) & \longmapsto & \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r). \\ r+\varepsilon v_{\pm 1}(θ,r). \end{array}\right), \end{eqnarray} where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions $$ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f…
Smooth surface systems may contain smooth curves which have no measure of maximal entropy
研究解析子集的最大熵测度，构造含无最大熵测度光滑曲线的曲面系统，并刻画一般及 $h$-扩张系统中相关子集的熵与 Hausdorff 维数性质。
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