Reversing Symmetry Groups of Cat Maps

方向反转与不可逆二维分段线性映射中的鲁棒混沌

Robust chaos in orientation-reversing and non-invertible two-dimensional piecewise-linear maps

本文将二维边界碰撞范式的鲁棒混沌构造从可逆保向情形推广到不可逆与方向反转情形，证明鲁棒混沌与映射全局拓扑性质无关，并确定了混沌区域及其分岔边界。

具有反向定向对称性的 MHD 平衡的闭合轨道

Closed orbits of MHD equilibria with orientation-reversing symmetry

本文研究具有非反射型反向定向对称性的 MHD 平衡，利用拓扑方法证明在压力函数具有环形嵌套水平集时环面上所有轨道均周期，并提出刻画微分同胚作用的指标。

A combinatorial model for reversible rational maps over finite fields

We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automorphism with a single family of reversing symmetries, a universal (i.e., map-independent) distribution function R(x)=1-e^{-x}(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit (J. A. G. Roberts and F. Vivaldi, Nonlinearity 18 (2005) 2171-2192). We show that these statistics correspond to those of a composition of two random involutions, having an appropriate number of f…

Resonances for rational Anosov maps on the torus

A complete description of resonances for rational toral Anosov diffeomorphisms preserving certain Reinhardt domains is presented. As a consequence it is shown that every homotopy class of two-dimensional Anosov diffeomorphisms contains maps with the sequence of resonances decaying stretched-exponentially. This is achieved by introducing a certain group of rational toral diffeomorphisms and computing the resonances of the respective composition operator considered on suitable anisotropic spaces of hyperfunctions. The class of examples is sufficiently rich to also include non-linear Anosov maps …

Poincaré maps and suspension flows: a categorical remark

Poincaré maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincaré maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the constructi…

Furstenberg Maps For CAT(0) Targets Of Finite Telescopic Dimension

We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with amenability and ergodicity properties, we prove the existence of equivariant maps from $B$ to the visual boundary $\partial X$.

Small scale quantum ergodicity in cat maps. II. Quasimodes that are not equidistributed at the logarithmical scales

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps'"). In Part II of the series, we construct quasimodes that are quantum ergodic but are not equidistributed at the logarithmical scales.

量子动力学从混沌中涌现：预量子猫映射的情况

Emergence of quantum dynamics from chaos: The case of prequantum cat maps

本文提出了一种新的量子化理论，通过研究预量子转移算子的共振，揭示了猫映射中量子动力学与混沌之间的关系。

Small scale quantum ergodicity in cat maps. I

In this series, we investigate quantum ergodicity at small scales for linear hyperbolic maps of the torus ("cat maps"). In Part I of the series, we prove quantum ergodicity at various scales. Let $N=1/h$, in which $h$ is the Planck constant. First, for all integers $N\in\mathbb{N}$, we show quantum ergodicity at logarithmical scales $|\log h|^{-α}$ for some $α>0$. Second, we show quantum ergodicity at polynomial scales $h^α$ for some $α>0$, in two special cases: $N\in S(\mathbb{N})$ of a full density subset $S(\mathbb{N})$ of integers and Hecke eigenbasis for all integers.

Stable mixing for cat maps and quasi-morphisms of the modular group

It is well-known that the action of a hyperbolic element (``cat map'') of the modular group on the 2-torus has strong chaotic dynamical properties such as mixing and exponential decay of correlations. In this note we study stability of this behaviour with respect to kicks. Our approach is based on geometric group theory, and in particular on a new result on quasimorphisms of the modular group.

关于高维猫映射的量子遍历性

On quantum ergodicity for higher dimensional cat maps

本文证明高维遍历猫映射在密度为一的逆 Planck 常数序列上本征函数均匀分布，并借助加性组合学与张量积结构工具拓展二维结果。

量子猫映射特征函数的等分布

Equidistribution of Eigenfunctions of Quantum Cat Maps

研究证明量子猫映射的短周期本征函数在 semiclassical 测度下等分布，并显示其大 $\ell^\infty$-范数在有限坐标上集中，证实了早期数值结果，与 Faure 等人的疤痕现象形成对比。