Rotation Numbers and Instability Sets

Generic Rotation Sets in Hyperbolic Surfaces

We show that for generic homeomorphisms homotopic to the identity in a closed and oriented surface of genus $g>1$, the rotation set is given by a union of at most $2^{5g-3}$ convex sets. Examples showing the sharpness for this asymptotic order are provided.

$\T$ 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。

Rotation Sets of Open Billiards

We investigate the rotation sets of open billiards in $\mathbb{R}^N$ for the natural observable related to a starting point of a given billiard trajectory. We prove that the general rotation set is convex and the set of all convex combinations of rotation vectors of periodic trajectory $P_ϕ$ is dense in it. We provide a constructive proof which illustrates that the set $P_ϕ$ is dense in the pointwise rotation set, and the closure of the pointwise rotation set is convex. We also consider a class of billiards consisting of three obstacles and construct a sequence in the symbol space such that it…

Rational Polygons as Rotation Sets of Generic Homeomorphisms of the Two-Torus

We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2).

Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle T. A set of integers is called r-Bohr if it is recurrent for all products of r rotations on T, and Bohr if it is recurrent for all products of rotations on T. It is a result due to Katznelson that for each r there exist sets of integers which are r-Bohr but not (r+1)-Bohr. We present new examples of r-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial pr…

Rotation Sets of Billiards with N Obstacles on a Torus

For billiards with $N$ obstacles on a torus, we study the behavior of specific kind of its trajectories, \emph{the so called admissible trajectories}. Using the methods developed in \cite{1}, we prove that the \emph{admissible rotation set} is convex, and the periodic trajectories of admissible type are dense in the admissible rotation set. In addition, we show that the admissible rotation set is a proper subset of the general rotation set.

Set-oriented numerical computation of rotation sets

We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of ε-rotation sets. These are obtained by replacing orbits with ε-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as ε decreases to zero. Based on this result, we prove the convergence of the numerical approximations as precision and iteration time tend to infinity. Further, we provide analytic error estimates for the algorithm under…

Bounded remainder sets for rotations on the adelic torus

In this paper we give an explicit construction of bounded remainder sets of all possible volumes, for any irrational rotation on the adelic torus $\mathbb A/\mathbb Q$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument which originated in the study of deformation properties of mathematical quasicrystals.

旋转与斜向辐照对趋光性藻类悬浮液不稳定性的影响

Rotation and Oblique Irradiation Effects on Phototactic Algal Suspension Instability

本文研究旋转与斜向辐照共同作用下趋光性藻类悬浮液的生物对流不稳定性，通过线性扰动理论分析发现旋转起稳定作用而斜向辐照起失稳作用。

Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets.