Non-degenerate quadratic laminations

Newhouse Laminations

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.

与二次映射相关的嵌铺与 Lyubich-Minsky 叶状结构，I：捏半共轭

Tessellation and Lyubich-Minsky laminations associated with quadratic maps, I: Pinching semiconjugacies

本文引入双曲和抛物二次映射填充 Julia 集的嵌铺，利用瓦片组织内部动力学，并基于嵌铺与线性化坐标退化研究构造了双曲到抛物退化的捏半共轭连续族。

Tessellation and Lyubich-Minsky laminations associated with quadratic maps II: Topological structures of 3-laminations

According to an analogy to quasi-Fuchsian groups, we investigate topological and combinatorial structures of Lyubich and Minsky's affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasi-Fuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to…

Collet-Eckmann 型条件与单临界二次层压的共形焊接

Collet-Eckmann type conditions and conformal welding of unicritical quadratic laminations

本文引入单临界层压的 Collet-Eckmann 型条件，证明其蕴含 Hölder 连续共形焊接并产生单临界多项式的 Julia 集，从而在不使用 Beurling 定理的情况下重新证明几乎所有角度对应具有 Hölder Fatou 分量的二次多项式。

Matings with laminations

We give a topological description of the space of quadratic rational maps with superattractive two-cycles: its "non-escape locus" M2 (the analog of the Mandelbrot set M) is locally connected, it is the continuous image of M under a canonical map, and it can be described as M (minus the 1/2-limb), mated with the lamination of the basilica. The latter statement is a refined version of a conjecture of Ben Wittner, which in its original version requires local connectivity of M to even be stated. Our methods of mating with a lamination also apply to dynamical matings of certain non-locally connecte…

The Heights Theorem for infinite Riemann surfaces

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel …

The combinatorial Mandelbrot set as the quotient of the space of geolaminations

We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $σ_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quo…

An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum

We construct an orientable holomorphic quadratic differential on a Riemann surface of genus 4 whose SL(2,R)-orbit is closed and has a highly degenerate Kontsevich - Zorich spectrum. This example is related to a previous similar construction in genus 3 by the first author.

The dual nest for degenerate Yoccoz puzzles

The Yoccoz puzzle is a fundamental tool in Holomorphic Dynamics. The original combinatorial argument by Yoccoz, based on the Branner-Hubbard tableau, counts the preimages of a non-degenerate annulus in the puzzle. However, in some important new applications of the puzzle (notably, matings of quadratic polynomials) there is no non-degenerate annulus. We develop a general combinatorial argument to handle this situation. It allows to derive corollaries, such as the local connectedness of the Julia set, for suitable families of rational maps.

Combinatorics, geometry and attractors of quasi-quadratic maps

The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $ω(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynam…

Viana映射推广的构建扩展

Building Expansion for Generalizations of Viana Maps

本文将Viana构造中底层的二次映射替换为具有更退化临界点的映射，从而扩展了具有两个正Lyapunov指数的映射例子。

Dynamics of an impact oscillator near a degenerate graze

We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate {\em chatter}. We also describe the dynamics associated with generic one-parameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from s…

二维复空间中的退化同宿分岔

Degenerate homoclinic bifurcations in complex dimension 2

研究二维复空间中同宿分岔，通过采用Takens论证证明展开非持久同宿相切的族含二次相切，结合相关成果得到复亨农映射分岔轨迹中鲁棒同宿相切丰富，还研究了由持久相切族引起的分岔，为复纽豪斯现象提供新途径。

The degenerate C. Neumann system I: symmetry reduction and convexity

The C. Neumann system describes a particle on the sphere S^n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has l+1 distinct eigenvalues with multiplicity. Each group of m_σequal eigenvalues gives rise to an O(m_σ)-symmetry in configuration space. The combined symmetry group G is a direct product of l+1 such factors, and its cotangent lift has an Ad^*-equivariant Momentum mapping. Regular reduction leads to the Rosochatius system on S^l, which has the same form as the Neumann system albeit for an additional effective potential. To underst…