The existence of invariant curves of a kind of almost periodic twist mappings

关于扭转映射台球解析族周期不变曲线的存在性

On the existence of periodic invariant curves for analytic families of twist maps and billiards

本文证明环面扭转映射解析族中给定旋转数的周期不变曲线要么对所有参数值存在，要么至多存在于离散子集中，并将此结论推广并应用于台球的有理焦散线模型。

Twist number and order properties of periodic orbits

A less studied numerical characteristic of periodic orbits of area preserving twist maps of the annulus is the twist or torsion number, called initially the amount of rotation. It measures the average rotation of tangent vectors under the action of the derivative of the map along that orbit, and characterizes the degree of complexity of the dynamics. The aim of this paper is to give new insights into the definition and properties of the twist number, and to relate its range to the order properties of periodic orbits. We derive an algorithm to deduce the exact value or a demi--unit interval con…

A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the new family of Lax-Oleinik type operators with an arbitrary $u\in C(M,\mathbb{R}^1)$ …

Uniform bounds for pre-periodic points in families of twists

Let phi be a morphism of projective N-space defined over a number field K. We prove that there is a bound B depending only on phi such that every twist of phi has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.

Existence and Stability of Almost Periodic Solutions of Differential Equations with Generalized Piecewise Constant Argument

This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.

双曲准周期不变环面的计算机辅助证明：耗散扭转映射

Computer assisted proofs for hyperbolic quasi-periodic invariant tori in dissipative twist maps

本文提出并实现了一种基于后验KAM型定理的计算机辅助方法，用于证明耗散标准映射中双曲准周期不变环面的存在性，并为特定参数提供了不变圆的严格证明。

$J^+$-like invariants of periodic orbits of the second kind in the restricted three body problem

We determine three invariants: Arnold's $J^+$-invariant as well as $\mathcal{J}_1$ and $\mathcal{J}_2$ invariants, which were introduced by Cieliebak-Frauenfelder-van Koert, of periodic orbits of the second kind near the heavier primary in the restricted three-body problem, provided that the mass ratio is sufficiently small.

XY模型的Aubry集与2-局部常数势的周期优化典型性

The Aubry set for the XY model and typicality of periodic optimization for 2-locally constant potentials

本文研究XY模型Aubry集的作用优化性质，给出依赖前两坐标势函数的最优周期测度显式表达，并利用弱KAM方法与变分技术证明了2-局部常数势周期优化的典型性。

Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles

The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions…

关于辛扭转映射族周期环面的脆弱性

On the fragility of periodic tori for families of symplectic twist maps

本文研究辛扭转映射族中Lagrange周期环面在势扰动下的脆弱性，证明非退化条件下指定旋转向量的周期环面仅能在有限参数值存在，并推出可积映射的刚性结果。

Chaotic dynamics in the Volterra predator-prey model via linked twist maps

We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator--prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.

Invariant graphs and spectral type of Schrödinger operators

In this paper we study spectral properties of Schrödinger operators with quasi-periodic potentials related to quasi-periodic action minimizing trajectories for analytic twist maps. We prove that the spectrum contains a component of absolutely continuous spectrum provided that the corresponding trajectory of the twist map belongs to an analytic invariant curve.

嵌入接触同调约束与空间等腰三体问题中的扭动动力学

ECH Constraints and Twist Dynamics in the Spatial Isosceles Three-Body Problem

研究通过嵌入接触同调约束揭示空间等腰三体问题中能量表面的动力学特性，证明临界水平以下能量表面存在无限多个周期轨道。