Discrete and Continuous Topological Dynamics: Fields of Cross Sections and Expansive Flows

N-expansive Flows

We define the concept of $N$-expansivity for flows and extend some of the results already established for discrete dynamics and for $CW$-expansive flows. We show examples of $N$-expansive flows but not expansive, and examples of $CW$-expansive flows but not $N$-expansive for any natural number $N$.} We also define Komuro $N$-expansivity and prove that on compact surfaces it implies Komuro expansivity.

Centralizers of hyperbolic and kinematic-expansive flows

We show generic $C^\infty$ hyperbolic flows (Axiom A and no cycles, but not transitive Anosov) commute with no $C^\infty$-diffeomorphism other than a time-t map of the flow itself. Kinematic expansivity, a substantial weakening of expansivity, implies that $C^0$ flows have quasi-discrete $C^0$-centralizer, and additional conditions broader than transitivity then give discrete $C^0$-centralizer. We also prove centralizer-rigidity: a diffeomorphism commuting with a generic hyperbolic flow is determined by its values on any open set.

On two notions of expansiveness for continuous semiflows

We study two notions of expansiveness for continuous semiflows: expansiveness in the sense of Alves, Carvalho and Siqueira (2017), and an adaptation of positive expansiveness in the sense of Artigue (2014). We prove that if $X$ is a metric space and $ϕ$ is an expansive semiflow on $X$ according to the first definition, then the semiflow $ϕ$ is trivial and the space $X$ is uniformly discrete. In particular, if $X$ is compact then it is finite. With respect to the second definition, we prove that if $X$ is a compact metric space and $ϕ$ is a positive expansive semiflow on it, then $X$ is a union…

Expansive algebraic actions of discrete residually finite amenable groups and their entropy

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index…

Anosov微分同胚的拓扑扩张性与阴影性

Topological Expansivity and Shadowing for Anosov Diffeomorphisms

本文证明Anosov微分同胚具有拓扑扩张性，并揭示Lipschitz阴影性与一致横截性及阴影常数的定量联系，同时给出两个反例说明扩张性与阴影性在完备黎曼流形上并非等价。

Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications

In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Our technique is also significant from a dynamical point of view, as it allows to detect complex dynamics. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivi…

渐近截面双曲流理论的若干贡献

Contributions to the Theory of Asymptotically Sectional Hyperbolic Flows

该研究针对渐近截面双曲（ASH）流理论展开，证明了星形ASH吸引子的截面双曲性、ASH吸引子的熵柔性等性质，并给出三维ASH吸引子周期轨道增长率下界，完善了相关动力系统理论。

无共轭点粗双曲流形上测地流最大熵测度的唯一性

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

本文在无共轭点且基本群为Gromov双曲、剩余有限的闭黎曼流形上，通过对扩张集施加适当条件，证明了测地流最大熵测度的唯一性，从而推广了此前仅在负截面曲率背景度量下成立的相关结果。

Symbolic dynamics and transfer operators for Weyl chamber flows: a class of examples

We provide special cross sections for the Weyl chamber flow on a sample class of Riemannian locally symmetric spaces of higher rank, namely the direct product spaces of Schottky surfaces. We further present multi-parameter transfer operator families for the discrete dynamical systems on Furstenberg boundary that are related to these cross sections.

Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems

There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map $Q$. Our first result gives a partial answer to a question of Lagarias and Wang: we char…

On free discrete subgroups of Diff(I)

We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.

LaSalle不变性原理在离散时间动力系统中的应用：一份简洁且自包含的教程

LaSalle Invariance Principle for Discrete-time Dynamical Systems: A Concise and Self-contained Tutorial

本文研究问题是如何在离散时间动力系统中应用LaSalle不变性原理；核心方法是自包含地整理了相关概念、引理和定理；主要贡献是提供了LaSalle不变性原理在离散时间系统中的证明和扩展。

离散群 $G$ 的自同构在 Sub$_G$ 上作用的动力学及其在 Lie 群格中的应用

Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups

本文研究了离散群自同构在子群空间上的动力学行为及连通 Lie 群中格的结构性质，证明了多循环群或 Lie 群格不存在膨胀自同构，并给出了自同构在 Lie 群及其格的子群空间上远距作用的等价条件。

离散动力学的有理解释器：$p$-adic域上的存在性、精确性与分解

Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields

本文研究了$p$-adic域上离散动力学的有理解释器的存在性、精确性与分解，提出动态中国剩余定理并探讨了逆极限扩展。