Renormalization of symmetric bimodal maps with low smoothness

圆周映射的重整化与 Arnold 舌的光滑性

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。

对称Fibonacci双峰映射存在野生吸引子

Wild attractor exists for symmetric Fibonacci bimodal map

本文证明对称Fibonacci双峰映射在临界阶足够大时，其临界点的公共ω-极限集为野生Cantor吸引子，并通过半共轭将问题简化为Fibonacci盒映射。

Chaotic Period Doubling

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…

Renormalization for piecewise smooth homeomorphisms on the circle

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy-Veech renormalizations of generalized interval exchanges maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converges to the set of piecewise affine interval exchange maps.

Explosion of smoothness for conjugacies between multimodal maps

Let $f$ and $g$ be smooth multimodal maps with no periodic attractors and no neutral points. If a topological conjugacy $h$ between $f$ and $g$ is $C^{1}$ at a point in the nearby expanding set of $f$, then $h$ is a smooth diffeomorphism in the basin of attraction of a renormalization interval of $f$. In particular, if $f:I \to I$ and $g:J \to J$ are $C^r$ unimodal maps and $h$ is $C^{1}$ at a boundary of $I$ then $h$ is $C^r$ in $I$.

Renormalization of $C^r$ Hénon map : Two dimensional embedded map in three dimension

We study renormalization of highly dissipative analytic three dimensional Hénon maps $$ F(x,y,z) = (f(x) - \varepsilon(x,y,z),\ x,\ δ(x,y,z)) $$ where $ \varepsilon(x,y,z) $ is a sufficiently small perturbation of $ \varepsilon_{2d}(x,y) $. Under certain conditions, $ C^r $ single invariant surfaces each of which is tangent to the invariant plane field over the critical Cantor set exist for $ 2 \leq r < \infty $. The $ C^r $ conjugation from an invariant surface to the $ xy- $plane defines renormalization two dimensional $ C^r $ Hénon-like map. It also defines two dimensional embedded $ C^r $ …