Uniform Periodic Point Growth in Entropy Rank One

$n$-环面上自同态的一致分布周期轨道

Uniformly distributed periodic orbits of endomorphisms on $n$-tori

本文证明了环面上任意遍历自同态存在度量意义下一致分布的周期轨道序列，并得出自同态遍历性等价于 Haar 测度可由周期测度逼近的推论。

Is uniform persisitence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson systems

Using techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general non-autonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.

Convergence of Periodically-Forced Rank-Type Equations

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.

Periodic Point Data Detects Subdynamics in Entropy Rank One

A framework for understanding the geometry of continuous actions of Z^d was developed by Boyle and Lind using the notion of expansive behavior along lower-dimensional subspaces. For algebraic Z^d-actions of entropy rank one, the expansive subdynamics is readily described in terms of Lyapunov exponents. Here we show that periodic point counts for elements of an entropy rank one action determine the expansive subdynamics. Moreover, the finer structure of the non-expansive set is visible in the topological and smooth structure of a set of functions associated to the periodic point data.

Accumulation of periodic points for local uniformly quasiregular mappings

We consider accumulation of periodic points in local uniformly quasiregular dynamics. Given a local uniformly quasiregular mapping $f$ with a countable and closed set of isolated essential singularities and their accumulation points on a closed Riemannian manifold, we show that points in the Julia set are accumulated by periodic points. If, in addition, the Fatou set is non-empty and connected, the accumulation is by periodic points in the Julia set itself. We also give sufficient conditions for the density of repelling periodic points.

A note on uniform exponential stability of linear periodic time-varying systems

In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix $A(t).$ Finally we analyze the robustness of the stability property under external disturbance.

鲁棒周期稳定性蕴含马尔可夫跳跃线性系统的均匀指数稳定性及随机线性常微分方程

Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations

本文证明线性cocycle鲁棒周期稳定性蕴含均匀稳定性，涉及马尔可夫跳跃线性系统和随机线性常微分方程的稳定性分析。

Model sets, almost periodic patterns, uniform density and linear maps

This article consists in two independent parts. In the first one, we investigate the geometric properties of almost periodicity of model sets (or cut-and-project sets, defined under the weakest hypotheses); in particular we show that they are almost periodic patterns and thus possess a uniform density. In the second part, we prove that the class of model sets and almost periodic patterns are stable under discretizations of linear maps.

Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy

We determine positive-dimensional G-periodic proper subvarieties of an n-dimensional normal projective variety X under the action of an abelian group G of maximal rank n-1 and of positive entropy. The motivation of the paper is to understand the obstruction for X to be G-equivariant birational to the quotient variety of an abelian variety modulo the action of a finite group.

Directional uniformities, periodic points, and entropy

Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar dynamical invariants for homeomorphisms, like entropy and periodic point data, become more complex and permit multiple definitions. We briefly survey some of these and other related invariants in the setting of algebraic $\mathbb{Z}^d$-actions, showing how, even in settings where the natural entropy as a $\mathbb{Z}^d$-action vanishes, a powerful theory of directio…

非一致耗散随机微分方程的随机周期解

Random periodic solutions for stochastic differential equations with non-uniform dissipativity

本文研究非一致耗散随机微分方程随机周期解的存在性与唯一性，通过反射耦合和同步耦合方法，分别处理无记忆及有限/无限时滞情形，证明了分布意义与路径wise意义下的解的存在性。

Uniform persistence criteria for a variable inputs chemostat model with delayed response in growth and complete analysis of the periodic case

We study a single-species chemostat model with variable nutrient input and variable dilution rate with delayed (fixed) response in growth. The first goal of this article is to prove that persistence implies uniform persistence. Then we concentrate in the particular case with periodic nutrient input and same periodic dilution with delayed response in growth. We obtain a threshold for either the (uniform) persistence of the model or that the biomass of every solution tends to vanish. Furthermore, we prove that persistence is equivalent to the existence of a unique non-trivial periodic solution. …

非均匀高阶格是特征刚性的

Non-uniform higher-rank lattices are character rigid

该研究证明了特征不为2的半单群中非均匀高阶不可约格的特征刚性，由此推出保概率测度作用的稳定子刚性与不变随机子群刚性，全面证实了Stuck和Zimmer的相关猜想。

Measure and cocycle rigidity for certain non-uniformly hyperbolic actions of higher rank abelian groups

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.

Kupka-Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a certain degree of freedom for the choice of the second one. For a non-countable set of choices, this leaf is not periodic and the diffeomorphism is Kupka-Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. As a consequence of our construction,…

Integrate-and-fire models with an almost periodic input function

We investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and $μ$-almost periodic functions. Special attention is paid to the properties of a firing map and its displacement, which give information about the spiking behaviour of the system under consideration. We provide conditions under which such maps are well-defined for every $t \in \mathbb R$ and are uniformly continuous. Moreover, we show that the LIF model with a Stepanov almost periodic input has a uniformly almost periodic displacement map. We also show that in the case of a $μ$-almost periodic drive it …

Uniform hyperbolicity revisited: Index of periodic points and equidimensional cycles

In this paper we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r >= 1). In the case of the C1-topology we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f have simple spectrum C1- robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of f are continuous in the wea…

Convergence in Comparable Almost Periodic Reaction-Diffusion Systems with Dirichlet Boundary Condition

The paper is to study the asymptotic dynamics in nonmonotone comparable almost periodic reaction-diffusion system with Dirichlet boundary condition, which is comparable with uniformly stable strongly order-preserving system. By appealing to the theory of skew-product semiflows, we obtain the asymptotic almost periodicity of uniformly stable solutions to the comparable reaction-diffusion system.

Maximum Rank and Asymptotic Rank of Finite Dynamical Systems

A finite dynamical system is a system of multivariate functions over a finite alphabet used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. The rank of a finite dynamical system is the cardinality of its image; the periodic rank is the number of its periodic points. In this paper, we determine the maximum rank and the maximum periodic rank of a finite dynami…

关于常规平面多项式仿射变换族的公共周期点的统一界

Uniform bound on common periodic points for families of regular plane polynomial automorphisms

研究了常规平面多项式仿射变换族中公共周期点的统一界问题，证明了在特定条件下公共周期点数量被统一常数限制。

Mixing on a class of rank one transformations

We prove that a rank one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences in the sense that the mean ergodic theorem holds for what we call dynamical sequences. In particular, Adams' class of staircase transformations and Ornstein's class constructed using ``random spacers'' have both restricted growth and uniformly ergodic spacer sequences.

Specification for Group Actions on Uniform Spaces

We extend specification and periodic specification to finitely generated group actions on uniform spaces using a concept of specification point. We prove that certain group actions having two distinct specification points have positive entropy. We further prove that if a group containing an infinite order element acts on an infinite Hausdorff uniform space and the action possesses periodic specification, then it is Devaney chaotic.

Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits

For a topological dynamical system $(X, T)$ we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal cardinality of uniform generators has lead us to new challenges in the theory of symbolic extensions. At the beginning we show that uniform generators can be identified with so-called symbolic extensions with an embedding, i.e., symbolic extensions admitting an equivariant measurable selector from preimages. From here we focus on such extensions and we …

The compound Poisson limit ruling periodic extreme behaviour of non-uniformly hyperbolic dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average si…

Uniform attractors for non-autonomous wave equations with nonlinear damping

We consider dynamical behavior of non-autonomous wave-type evolutionary equations with nonlinear damping, critical nonlinearity, and time-dependent external forcing which is translation bounded but not translation compact (i.e., external forcing is not necessarily time-periodic, quasi-periodic or almost periodic). A sufficient and necessary condition for the existence of uniform attractors is established using the concept of uniform asymptotic compactness. The required compactness for the existence of uniform attractors is then fulfilled by some new a priori estimates for concrete wave type eq…

关于动力系统一致遍历性的注记

A Note on the Uniform Ergodicity of Dynamical Systems

本文研究了不可逆拓扑与保测动力系统的一致遍历性，证明其分别等价于最终周期性和周期性，并通过对格同态长期行为的分析得出结论。

介质代数上的内在均匀结构

Intrinsic uniform structure on median algebras

本文介绍了介质代数上的内在预紧凸均匀结构，该结构在自然假设下为Hausdorff空间，并通过均匀完成得到最小介质紧致化。诱导拓扑提供了线性序集区间拓扑的高维类比。