Daniel Smania
We show the hyperbolicity of the Feigenbaum fixed point using the inflexibility of the Feigenbaum tower, the Mane-Sad-Sullivan $λ$-Lemma and the existence of parabolic domains (petals) for semi-attractive fixed points.
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We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and "hairiness" of the Mandelbrot set near the corresponding parameter values. We also conclude that the set of real infinitely renormalizable quadratics of type bounded by some N>1 has Hausdorff dimension strictly between 0 and 1. In the course of getting these results…
We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.
Limit Drift for Complex Feigenbaum Mappings
本文研究复 Feigenbaum 映射中塔动力学的漂移极限，证明了绝对连续不变测度的存在与收敛，并给出漂移的有限极限表达式。
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).
Time series are proficiently converted into graphs via the horizontal visibility (HV) algorithm, which prompts interest in its capability for capturing the nature of different classes of series in a network context. We have recently shown [1] that dynamical systems can be studied from a novel perspective via the use of this method. Specifically, the period-doubling and band-splitting attractor cascades that characterize unimodal maps transform into families of graphs that turn out to be independent of map nonlinearity or other particulars. Here we provide an in depth description of the HV trea…
The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which is also hyperbolic among generic smooth enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicit…
Validated enclosure of renormalization fixed points via Chebyshev series and the DFT
本文提出基于切比雪夫级数离散化的计算框架，证明多阶Feigenbaum-Cvitanović重整化算子不动点的存在性与稳定性，并高精度计算普适常数。
We explore fundamental questions about the renormalization group through a detailed re-examination of Feigenbaum's period doubling route to chaos. In the space of one-humped maps, the renormalization group characterizes the behavior near any critical point by the behavior near the fixed point. We show that this fixed point is far from unique, and characterize a submanifold of fixed points of alternative RG transformations. We build on this framework to systematically distinguish and analyze the allowed singular and `gauge' (analytic and redundant) corrections to scaling, explaining numerical r…
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。
Correspondences on Riemann surfaces and non-uniform hyperbolicity
本文证明 Riemann 曲面上的对应关系具有弱双曲性，并应用于有理映射：对具有 4 个后临界点的映射，存在有限的不变曲线或图同伦类集合，所有曲线或图在迭代提升下被吸引，从而提供拓扑标准形。
Saddle-point structure of fixed points in a reaction-diffusion equation with discontinuous nonlinearity
该研究针对含不连续非线性项的反应扩散方程，通过引入含Dirac delta分布的线性化方法分析不动点附近解的局部行为，证明定常解具鞍点结构并确立不动点的双曲性。
Area-preserving maps have been observed to undergo a universal period-doubling cascade, analogous to the famous Feigenbaum-Coullet-Tresser period doubling cascade in one-dimensional dynamics. A renormalization approach has been used by Eckmann, Koch and Wittwer in a computer-assisted proof of existence of a conservative renormalization fixed point. Furthermore, it has been shown by Gaidashev, Johnson and Martens that infinitely renormalizable maps in a neighborhood of this fixed point admit invariant Cantor sets with vanishing Lyapunov exponents on which dynamics for any two maps is smoothly c…
Discontinuous Generalized Hyperbolicity and Structural Stability
本文利用Michael连续选择定理，针对不连续广义双曲分裂的Banach空间动力学，建立了上同调方程的有界连续解，证明了广义双曲微分同胚在扰动下的半共轭性，并对一维Morse-Smale系统的无限乘积获得了无扩张性的完全结构稳定性。
We examine thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the Thue-Morse substitution, and in complex dynamics by the Feigenbaum-Coullet-Tresser map. The basic question answered is whether fixed points $V$ of a renormalization operator $\CR$ acting on the space of potentials are such that the pressure function $γ\mapsto \CP(-γV)$ exhibits phase transitions. This extends the work by Baraviera, Leplaideur and Lopes on the Manneville-Pomeau map, where such phase transitions were indeed detected. In this paper, however, the attra…
Thermodynamic formalism for intermittent maps with multiple neutral fixed points and phase transitions
本文针对具有多个中性不动点的间歇映射与不连续势函数发展热力学形式主义，证明满足双曲性假设时存在具正熵且关联指数衰减的唯一平衡态，还刻画了单参数族的相变并证明临界参数处多平衡态可共存，方法基于共形测度分析。
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