Are generic dynamical properties stable under composition with rotations?

$\T$ 同胚随机合成的旋转集

Rotation sets for random compositions of $\T$ homeomorphisms

本文以受经典工作启发的新型旋转集定义为工具，研究了移位空间上恒等同痕类中的 $\T$ 同胚 cocycle，探讨了不同概念的旋转集，并得出了关于旋转集形状、连续性及有界位移的初步结果与示例。

Ergodic Properties of Compositions of Interval Exchange Maps and Rotations

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T, there is a full measure set of αin [0, 1) so that T composed with R_α is uniquely ergodic, where R_α is rotation by α.

关于某些旋转线段的稳定与不稳定行为

On Stable and Unstable Behaviours of Certain Rotation Segments

该研究针对二维环面上旋转集为非平凡线段的非游荡同胚，证明其非通有性，刻画了保面积条件下的有界偏差与C^0扰动稳定性，同时提出了使旋转集被邻近系统吞噬的扰动方案，揭示了这类旋转集的稳定与不稳定特性。

Stability of the rotation set of area-preserving toral homeomorphisms

We show that if the rotation set of a homeomorphism of the torus is stable under small perturbations of the dynamics, then it is a convex polygon with rational vertices. We also show that such homeomorphisms are $C^0$-generic and have bounded rotational deviations (even for pseudo-orbits). The results hold both in the area-preserving setting and in the general setting. When the rotation set is stable, we give explicit estimates on the type of rationals that may appear as vertices of rotation sets in terms of the stability constants.

Approach to a rational rotation number in a piecewise isometric system

We study a parametric family of piecewise rotations of the torus, in the limit in which the rotation number approaches the rational value 1/4. There is a region of positive measure where the discontinuity set becomes dense in the limit; we prove that in this region the area occupied by stable periodic orbits remains positive. The main device is the construction of an induced map on a domain with vanishing measure; this map is the product of two involutions, and each involution preserves all its atoms. Dynamically, the composition of these involutions represents linking together two sector maps…

Rotation numbers for planar attractors of equivariant homeomorphisms

Given an integer m>1 we consider Zm-equivariant and orientation preserving homeomorphisms in the plane with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated dynamical features. The key is a preliminary construction of Zm-equivariant Denjoy maps of the circle.