A note on the relation between fixed point and orbit count sequences
Michael Baake, Natascha Neumaerker
从算术函数空间上线性映射的角度研究了不动点与轨道计数序列之间的关系。谱和渐近性质的推导和几个数量明确给出的高斯二项式系数。
The relation between fixed point and orbit count sequences is investigated from the point of view of linear mappings on the space of arithmetic functions. Spectral and asymptotic properties are derived and several quantities are explicitly given in terms of Gaussian binomial coefficients.
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A Note on Homoclinic Orbits for Second Order Hamiltonian Systems
本文研究二阶哈密顿系统的同宿轨道的存在性。在适当的条件下，利用直接变分法和傅里叶分析，证明了在位势$V$上存在同宿轨道。
A note on the density of periodic orbits of Anosov geodesic flow in manifolds of finite volume
该研究证明有限体积流形上Anosov测地流的周期轨道具有稠密性，且该结果可推广至非紧场景的保守Anosov流，丰富了动力系统相关理论。
A note on the bounded orbit conjecture
本文证明：平面上无不动点、反定向且所有轨道有界的同胚必有无穷多个周期轨道，从而为有界轨道猜想提供了一个注记。
A note on Zariski dense orbit conjecture
本文给出Zariski稠密轨道猜想的一个注记，简化了射影三维簇自同构的ZDO结果，并证明了不规则度足够大的射影簇的ZDO成立。
Note on the points with dense orbit under $\times 2$ and $\times 3$ maps
Furstenberg证明了对任何$x\in [0,1]\backslash Q$，$$ \dim_H \bar{\{2^nx ({\text{mod}}\ 1): n\ge 1\}}+ \dim_H \bar{\{3^nx ({\text{mod}}\ 1): n\ge 1\}}\ge 1. $$当$x$是一个正规数时，上述结果平凡地成立。本文给出了上述维数公式成立的显式非正规数。
A note on odometers and shadowing
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Functorial orbit counting
研究了映射的闭轨道的函子性质和增长性质。通过将任意序列看作映射的轨道计数函数，映射的迭代和笛卡儿积定义了整数序列之间的新变换。轨道幺半群与任何整数序列相关联，给出了欧拉变换的动力学解释。
A note on the periodic orbits of Wolbachia spread dynamics in mosquito populations in periodic environments
本文反驳了周期环境中Wolbachia传播模型的周期轨道数量猜想，并重建证明：任意周期映射序列的非零周期轨迹数不超过两个。
A note on integrability and finite orbits for subgroups of Diff(C^n,0)
本文将Mattei-Moussu和Camara-Scardua在2维空间中的两个结果推广到任意维空间。我们还提供了奇异叶理具有西格尔型奇点的例子，并回答了卡马拉-斯卡杜阿以前工作中悬而未决的中心问题。
These notes were written during the 9th and 10th sessions of the subject Dynamical Systems II coursed at DTU (Denmark) during the Winter Semester 2015-2016, and later extended in February 2017. They aim to provide students with a theoretical and numerical background for the computation of periodic orbits using Newton's method. We focus on periodically perturbed quasi-integrable systems (using the forced pendulum as an example) and hence we take advantage of the Melnikov method to get first guesses. However, these well known techniques are general enough to be applied in other type of systems. …
We show that a self orbit equivalence of a transitive Anosov flow on a $3$-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is $\mathbb{R}$-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is $3$-dimensional.
These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of finding periodic orbits of prescribed energy for autonomous Tonelli Lagrangian systems on the twisted cotangent bundle of a closed manifold. In the first part of the lecture notes, we put together in a general theorem old and new results on the subject. In the second part, we focus on an important class of examples: magnetic flows on surfaces. For such system…
It is shown that the restriction of the action of any group with finite orbit on the minimal sets of dendrites is equicontinuous. Consequently, we obtain that the action of any amenable group and Thompson group on dendrite restricted on minimal sets is equicontinuous. We further provide a class of non-amenable groups whose action on dendrites has finite orbit. We extend also some of our results to dendron. We further give a characterization of the set of invariant probability measures and its extreme points.
Counting points by height in semigroup orbits
针对极化动力系统半群轨道中高度有界点的计数问题，本文利用 Wiener-Ikerhara Tauberian 定理改进已有估计，并得到射影直线上一般半群的精确渐近式。
Let $f_{c,d}(x)=x^d+c\in \mathbb{C}[x]$. The $c_0$ values for which $f_{c_0,d}$ has a strictly pre-periodic finite critical orbit are called Misiurewicz points. Any Misiurewicz point lies in $\bar{\mathbb{Q}}$. Suppose that the Misiurewicz points $c_0,c_1\in \bar{\mathbb{Q}}$ are such that the polynomials $f_{c_0,d}$ and $f_{c_1,d}$ have the same orbit type. One classical question is whether $c_0$ and $c_1$ need to be Galois conjugates or not. Recently there has been a partial progress on this question by several authors. In this note, we prove some new results when $d$ is a prime. All the res…
Given a set of endomorphisms on $\mathbb{P}^N$, we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.
A short note on the orbit growth of sofic shifts
本文通过Artin-Mazur zeta函数及最小右解析表示，简短证明了sofic移位素轨道与Mertens轨道计数函数的渐近行为。
The acclaimed merits of analytical solutions based on a fictitious time developed in the 1970's were partially overvalued due to a common misuse of classical analytical solutions based on the physical time that were taken as reference. With the main problem of the artificial satellite theory as a model, we carry out a more objective comparison of both kinds of theories. We find that the proper initialization of classical solutions notably balances the performance of the two distinct approaches in what respects to accuracy. Besides, extension of both kinds of satellite theories to higher orders…
For an $α$-expansive homeomorphism of a compact space we give an elementary proof of the following well-known result in topological dynamics: A sufficient condition for the homeomorphism to have the shadowing property is that it has the $α$-shadowing property for one-jump pseudo orbits (known as the local product structure property). The proof relies on a reformulation of the property of expansiveness in terms of the pseudo orbits of the system.
A note on Reeb dynamics on the tight 3-sphere
我们证明了3-球面上的非退化紧切触形式恰好有两个简单闭Reeb轨道当且仅当线性化切触同调中的微分为零。此外，在这种情况下的Floquet乘子和Conley-Zehnder指数的两个Reeb轨道同意在4-空间中的一个合适的无理椭球。
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在本文中，我们考虑以下两个家庭的二次多项式$S_{a,c}(x) = a - cx^2$和$T_{a,c}(x) = a - c(1 + x^2)$，并表明他们各自的周期3轨道生活非常不同的生活。
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本文证明了在双曲空间中的n维正则单形内存在一个（n+1）周期台球轨迹，它与每个面的内部恰好碰撞一次.
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本文给出了一个具有两个连通分支和一个完备度量的一维流形的例子，该流形的等距群有一个不闭的轨道。这回答了S. Gao和A. S.凯克里斯
These are lecture notes on Floer and Rabinowitz-Floer homology written for a graduate course at UNICAMP August-December 2016 and a mini-course held at IMPA in August 2017.
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