Michael Yampolsky
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
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Hyperbolicity of Renormalization for bi-cubic circle maps with bounded combinatorics
本文为具有有界组合的双三次圆映射构造了带有余维二稳定叶层的重正化双曲吸引子。
Hyperbolicity of renormalization of circle maps with a break-type singularity
研究了具有断点的圆同胚的重正化算子，证明了它具有双曲马蹄吸引子。
We prove that if two analytic multicritical circle maps with the same bounded type rotation number are topologically conjugate by a conjugacy which matches the critical points of the two maps while preserving the orders of their criticalities, then the conjugacy necessarily has $C^{1+α}$ regularity, where $α$ depends only on the bound on the type of the rotation number. We then extend this rigidity result to $C^3$-smooth bi-cubic circle maps.
Renormalization of unicritical analytic circle maps
本文通过证明具有三次临界点的解析临界圆映射的重整化的拟共形刚性，将其重整化理论推广到具有任意奇临界指数的映射的情形。
The Attractor of Renormalization and Rigidity of Towers of Critical Circle Maps
证明了临界圆映射重整化的一个具有康托尔集结构的整体吸引子A的存在性。集合A在广义重整化变换下是不变的，它对A的作用与双侧移位共轭。
Rigidity, universality,and hyperbolicity of renormalization for critical circle maps with non-integer exponents
本文构造了一个作用于解析圆映射的重整化算子，它的临界指数$α$不一定是奇数$2n+1$，$n\in\mathbb N$ .当$α=2n+1$时，我们的定义推广了解析临界圆映射的柱面重整化.在$α$接近于奇数的情况下，证明了有界型映射重整化的双曲性。我们用它来证明这类映射的普适性和$C^{1+α}$ -刚性。
Renormalization of circle maps and smoothness of Arnold tongues
本文研究重整化算子在包含三次临界圆周映射与圆周微分同胚的 Banach 流形上的全局行为，并据此证明了无理 Arnold 舌的光滑性。
Renormalization for piecewise smooth homeomorphisms on the circle
本文研究了作用在圆上分段光滑同胚上的重整化算子，其实质是研究亏格为1的广义区间交换映射的Rauzy-Veech重整化.特别是，我们证明了零平均非线性和满足一定的光滑性和组合假设的重正化的映射收敛到一组分段仿射区间交换映射。
On the convergence of renormalizations of piecewise smooth homeomorphisms on the circle
研究了圆上分片光滑同胚的重正规化，将这类映射看作亏格为1的广义区间交换映射。设$Df$在每个连续区间上绝对连续，且对于某些$p>1$，$D\ln{Df}\in \mathbb{L}_{p}$.证明了在一定的组合假设下，$f_{1}$和$f_{2}$的相应重正化在$C^{1+L_{1}}$ -范数下相互逼近.
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper we establish this principle for a large class of bicritical circle maps, which are $C^3$ circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo for the case o…
In this paper we give a new prove of hyperbolicity of renormalization of critical circle maps using the formalism of almost-commuting pairs. We extend renormalization to two-dimensional dissipative maps of the annulus which are small perturbations of one-dimensional critical circle maps. Finally, we demontsrate that a two-dimensional map which lies in the stable set of the renormalization operator possesses an attractor which is topologically a circle. Such a circle is critical: the dynamics on it is topologically, but not smoothly, conjugate to a rigid rotation.
Renormalization of circle diffeomorphisms with a break-type singularity
设$f$是一个具有无理旋转数的保向圆同态，它有一个断点$ξ_{0},$，即它的导数$f'$在此点有一个跳跃间断.假设$f'$满足依赖于参数$γ>0.$的Zygmund条件，证明了$f$的重整化在$C^{1}$ -范数下可用Möbius变换逼近，如果$γ\in (0,1]$，则它们在$C^{2}$ -范数下可用Möbius变换逼近，如果$γ\in (1,+\infty).$.
On the renormalizations of circle homeomorphisms with several break points
设$f$是圆上具有多个断点的方向保持同胚，即它的导数$Df$在这些断点处具有跳跃间断.研究了分片光滑圆同胚的Rauzy-Veech重正规化，将这类映射看作亏格为1的广义区间交换映射。设$Df$在每个连续区间上绝对连续，且对于某个$p>1$，$D\ln{Df}\in \mathbb{L}_{p}$是绝对连续的.证明了在一定的组合假设下，重整化$f$在$C^{1+L_{1}}$ -范数下可用分段Möbus函数逼近，即$R^{n}(f)$在$C^{1}$ -范数下逼近，$D^{2}R^{n}(f)$在$L_{1}$ -范数下逼近.特别地，如果$f$有平凡的产品的大小的休息，然后重新正规化近似分段仿射区间交换映射。
Simultaneous renormalization of commuting circle diffeomorphisms
本文为交换圆微分同胚对定义重整化算子，在加速Brun算法驱动下，证明对有界型旋转向量指数快速收敛到刚性旋转，并推广到满测度类，同时证明算子拓扑双曲性。
Rigidity of critical circle mappings, II
证明了任意两个具有三次临界点且无理旋转数相同的有界型实解析临界圆映射对于某些$α>0$是$C^{1+α}$共轭的.
Circle packings, renormalizations and subdivision rules
本文利用 Teichmüller 空间上剥皮映射的迭代，为满足特定细分规则的圆填充发展了重整化理论，证明重整化算子一致收缩，从而完整解决了此类圆填充的存在性与模问题，并证明了其几何不灵活性与组合刚性。
Asymptotic rigidity of scaling ratios for critical circle mappings
在本文中，我们建立了$C^2$的先验界的临界圆映射的比例的形式，也给出了一个紧性的重整化算子。
Invariant circles and phase portraits of cubic vector fields on the sphere
本文分类球面上的三次多项式向量场，识别 Kolmogorov 系统，证明完全可积情形存在，并给出齐次三次向量场各类不变圆数量的紧界及相关相图。
Immediate renormalization of cubic complex polynomials with empty rational lamination
本文研究三次复多项式的立即重整化性质，证明在Julia集无（预）周期割点的条件下，不属于不变填充Julia集的唯一临界点必为回归点。
Parabolic and near-parabolic renormalizations for local degree three
本文构造了由局部三次临界点全纯映射组成且在抛物与近抛物重整化下不变的新类，从而可将二次情形结果推广到三次单临界多项式。
Complex bounds for critical circle maps
我们使用的方法与M。Lyubich证明了实二次型的复界，推广了E.具有无理旋转数的临界圆映射的De Faria复先验界.临界圆映射的重正化的收缩性质如下。在附录中，我们给出了复界在证明某些朱利亚集局部连通性中的应用。
$C^1$ Robust Rigidity for Bi-critical Circle Maps
该研究针对双临界圆周映射的刚性问题，证明满足符号相同、重正化在$C^2$拓扑下指数共同收敛的拓扑共轭双临界圆周映射具有$C^1$共轭性，推进了一维动力系统刚性理论研究。
Complex a priori bounds for multicritical circle maps with bounded type rotation number
本文研究了具有多个临界点和有界型旋转数的圆同胚，证明了其复先验界，并得出具有相同有界型旋转数的双三次圆映射具有 $C^{1+α}$ 刚性的结论。
The rigidity problem for analytic critical circle maps
证明了若$f$和$g$是任意两个解析临界圆映射，且无理旋转数相同，则$f$的临界点映射到$g$的临界点的共轭在临界点处具有正则性$C^{1+α}$，其普适值为$α>0$ .由此，给出了临界圆映射的完全重整化马蹄的双曲性的一个新证明。
In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.
Rigidity of critical circle mappings, I
证明了两个具有相同旋转数的有界型$C^r$临界圆映射对于某些$α>0$是$C^{1+α}$共轭的，只要它们的连续重正化在$C^0$意义下以指数速度收敛在一起.数字$α$仅取决于收敛的收敛速率。我们还给出了具有相同旋转数的$C^\infty$临界圆映射的例子，这些映射对于任何$β>0$都不是$C^{1+β}$共轭的。
Hyperbolicity of renormalization of critical quasicircle maps
本文研究了临界准圆映射的重正化超几何性，扩展了Yampolsky的结果，通过冠状重正化算子证明了其超几何固定点性质。
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.
We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.
Laplacians on Julia Sets III: Cubic Julia Sets and Formal Matings
我们继续研究构造朱利亚集上的不变Laplacian，并研究其谱的性质。本文主要讨论两类例子：1）三次多项式$z^3 + c$的单临界点的朱利亚集; 2）二次朱利亚集的形式匹配.在本系列的前几篇文章中介绍的一般方案涉及将Julia集实现为具有标识的圆，并试图获得Laplacian作为从具有标识的圆模型导出的图上的图Laplacian的重正化极限。在三次朱利亚集的情况下，细节遵循为二次朱利亚集建立的模式，但对于配对，细节是相当具有挑战性的，我们只在一个例子中完全成功。一旦我们构造了拉普拉斯算子，我们就能够使用数值方法来近似本征值和本征函数。一个引人注目的观察数据是，三次朱利亚集的重数的所有特征空间（除了平凡的本征空间的常数）是偶数。对于前面研究过的二次朱利亚集，这样的情况是不成立的。我们能够解释这一点，是基于3是奇数这一事实，更确切地说，是因为二面角3对称群只有两个不同的一维不可约表示。
Rigidity for piece-wise smooth circle maps and certain GIETs
本文证明具有一般旋转数、均值为零非线性的分段C^3广义区间交换变换在边界等价且重整化趋近仿射情形下，其共轭映射为C^1，并推广至旋转型组合数据及断点等价的圆微分同胚情形。
We consider order preserving $C^3$ circle maps with a flat piece, Fibonacci rotation number, critical exponents $(\ell_1, \ell_2)$ and negative shwarzian derivative. This paper treat the geometry characteristic of the non-wondering (cantor (fractal)) set from a map of our class. We prove that, for $(\ell_1, \ell_2)$ in $(1,2)^2$, the geometry of system is degenerate (double exponentially fast). As consequences, the renormalization diverges and the geometric (rigidity) class depends on the three couples $(c_u(f), c'_u(f) )$, $( c_+(f), c'_+(f))$ and $(c_s(f), c'_s(f) )$.\vspace{0.5cm}
Hyperbolicity of renormalization for maps with multiple breaks
本文在断点小且旋转数有界类型的条件下，为多重断点圆周映射构造双曲马蹄，并证明其刚性类为解析子流形。
Instability of Renormalization
在经典动力系统的重整化理论中，如单峰映射和临界圆映射，拓扑共轭类是重整化的稳定流形。另一方面，物理上更现实的系统可能会在拓扑类中表现出重整化的不稳定性。这种不稳定性产生了新的现象，并开辟了超越经典理论的研究方向。在相空间中，它会导致共存现象，即存在吸引子有界的系统，而吸引子有退化的系统拓扑共轭;在参数空间中，它会导致维数不一致，即拓扑满族维数太少，无法实现所有可能的几何行为。
We describe some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. Analogous results are believed to be true for the self-d…
A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be \emph{immediately renormalizable} if there exists a (connected) quadratic-like invariant filled Julia set $K^*$ such that $b\in K^*$. In that case exactly one critical point of $P$ does not belong to $K^*$. We show that if, in addition, the Julia set of $P$ has no (pre)periodic cutpoints then this critical point is recurrent.
Decay of geometry for a class of cubic polynomials
该研究针对具有广义 Fibonacci 组合结构的双峰三次多项式，证明其双主巢缩放因子至少指数衰减，并据此得出不存在 Cantor 吸引子的结论。
Wandering intervals in affine extensions of self-similar interval exchange maps: the cubic Arnoux-Yoccoz map
本文利用重正化矩阵的模大于大于1的复特征值的代数性质和由自相似性产生的自然替换所建立的复分形，给出了具有游荡区间的仿射区间交换映射存在与之半共轭的充分条件.我们证明了三次Arnoux-Yoccoz区间交换映射满足这些条件。
A piecewise linear homeomorphism of the circle which is periodic under renormalization
本研究构造了一个斜率为2/3幂次、旋转数为√2-1且重整化后周期的圆周分段线性同胚，否定了Calegari的相关问题，证明F_{2/3}无法嵌入F，并给出其变体问题的肯定性证据与计算观测结果。
Renormalization and Central Limit Theorem for Critical Dynamical Systems with Weak External Noise
我们研究了弱噪声对临界一维映射的影响，即具有重整化理论的映射。本文建立了弱噪声下的一维中心极限定理，并得到了该收敛速度的Berry-Esseen估计.我们详细分析了倍周期累积时的映射和具有黄金平均旋转数的临界圆映射。利用重整化群方法，我们推导了长时间后的有效噪声的几个特征的标度关系。我们使用这些比例关系表明，弱噪声的中心极限定理在这两个例子中成立。我们注意到，对于这里给出的结果，映射具有抛物行为是必不可少的。它们对于双曲轨道是假的。
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