Dense set of large periodic points

Periodic points and transitivity on dendrites

We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition (we provide an example of a dynamical system on a dendrite with dense endpoints satisfying this assumption). We also show that it may happen that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It may also happen that dynamical system is transitive but there is a unique periodic point, which in fact …

三角台球中非周期且非处处稠密的轨迹

Non-periodic not everywhere dense trajectories in triangular billiards

利用诱导映射与区间交换变换，构造等腰三角形单参数族，其非周期轨迹非处处稠密，从而否定性解决三角台球中长期未决的稠密性问题。

可数马尔可夫移位的密集周期优化 via Aubry 点

Dense periodic optimization for countable Markov shift via Aubry points

本文研究了可数马尔可夫移位中密集周期优化问题，通过Aubry点概念证明了局部Hölder连续势函数的最大测度存在性。

Some remarks on periodic billiard orbits in rational polygons

A polygon is called rational if the angle between each pair of sides is a rational multiple of $π.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of the billiard flow. This is a strengthening of Masur's theorem, who has shown that any rational polygon has ``many'' periodic billiard trajectories; more precisely, the set of directions of the periodic trajectories are dense in the set of velocity directions $§^1.$ We will also prove some refinements of Theorem 1: the ``well distribution'' of periodic orbits in…

Monotone dynamical systems with dense periodic points

In this paper we prove a recent conjecture by M. Hirsch, which says that if $(f,Ω)$ is a discrete time monotone dynamical system, with $f\colon Ω\toΩ$ a homeomorphism on an open connected subset of a finite dimensional vector space, and the periodic points of $f$ are dense in $Ω$, then $f$ is periodic.

Periodic billiard orbits in right triangles II

Periodic billiard orbits are dense in the phase space of an irrational right triangle. A stronger pointwise density result is also proven.

具有稠密周期性的区间映射

Interval maps with dense periodicity

本文研究稠密周期点区间映射空间 CP 及其闭包 Cl(CP)，刻画其链回归特征、拓扑精确性、共轭类丰富性及 generic 映射的周期点结构与跟踪性质。

Dense existence of periodic Reeb orbits and ECH spectral invariants

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

Absolutely Periodic Billiard Orbits of Arbitrarily High Order

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is als…

Periodic points of birational maps on projective surfaces

We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.

Periodic points in the intersection of immediate attracting basins boundaries

We give sufficient conditions under which the set of eventually periodic points in the intersection of immediate attracting basins boundaries is non-empty. We give other conditions under which this set is dense in the intersection.

Random product of quasi-periodic cocycles

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of $C^r$, $0\leq r \leq \infty$ (or analytic) $k+1$-tuples of quasi periodic cocycles taking values in $SL_2(\mathbb{R})$ such that the random product of them has positive Lyapunov exponent contains a $C^0$ open and $C^r$ dense subset which is formed by $C^0$ continuity point of the Lyapunov exponent For $k+1$-tuples of quasi periodic cocycles taking values in $GL_d(\mathbb{R})$ for $d>2$, we prove that if one of them is diag…

Sensitive dependence and dense perodic points

We show sensitive dependece on initial condition and dense periodic points imply asymptotic sensitivity, a stronger form of sensitivity, where the deviation happens not just once but infintely many times. As a consequence it follows that all Devaney chaotic systems (e.g. logistic map) have this property.

Periodic billiard orbits in right triangle

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

关于有限体积流形上Anosov测地流周期轨道稠密性的注记

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。

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Consider a compact manifold M of dimension at least 2 and the space of C^r-smooth diffeomorphisms Diff^r(M). The classical Artin-Mazur theorem says that for a dense subset D of Diff^r(M) the number of isolated periodic points grows at most exponentially fast (call it the A-M property). We extend this result and prove that diffeomorphisms having only hyperbolic periodic points with the A-M property are dense in Diff^r(M). Our proof of this result is much simpler than the original proof of Artin-Mazur. The second main result is that the A-M property is not (Baire) generic. Moreover, in a Newhous…

元胞自动机的严格周期点与周期因子

Strictly periodic points and periodic factors of cellular automata

本文证明了具有几乎等度连续点的元胞自动机的严格时间周期点集在测度的拓扑支撑中稠密，扩展了 Lena 等人关于等度连续情形的结果。

Periodic points of algebraic actions of discrete groups

Let $Γ$ be a countable group. A $Γ$-action on a compact abelian group $X$ by continuous automorphisms of $X$ is called Noetherian if the dual of $X$ is Noetherian as a ${\mathbb Z}(Γ)$-module. We prove that any Noetherian action of a finitely generated virtually nilpotent group has a dense set of periodic points.

Lyapunov optimizing measures and periodic measures for $C^2$ expanding maps

We prove that there exists an open and dense subset $\mathcal{U}$ in the space of $C^{2}$ expanding self-maps of the circle $\mathbb{T}$ such that the Lyapunov minimizing measures of any $T\in{\mathcal U}$ are uniquely supported on a periodic orbit.This answers a conjecture of Jenkinson-Morris in the $C^2$ topology.

Hyperbolic periodic points for chain hyperbolic homoclinic classes

In this paper we establish a closing property and a hyperbolic closing property for thin trapped chain hyperbolic homoclinic classes with one dimensional center in partial hyperbolicity setting. Taking advantage of theses properties, we prove that the growth rate of the number of hyperbolic periodic points is equal to the topological entropy. We also obtain that the hyperbolic periodic measures are dense in the space of invariant measures.

Periodic geodesics on translation surfaces

As shown by Masur in 80s, for any translation surface there exists a periodic geodesic of bounded length, the directions of periodic geodesics are dense in the unit circle, and the number of cylinders of periodic geodesics of length at most T admits upper and lower quadratic estimates. The main goal of this paper is to generalize the above results, with special interest in obtaining effective estimates. In addition, we apply results of Eskin and Masur to establish some properties of periodic geodesics on generic translation surfaces.

Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, then periodic points in the boundary of A are dense in this boundary. To prove this in the non simply- connected or parabolic situations we prove a more abstract, geometric coding trees version.

Inverse periodic shadowing properties

We consider inverse periodic shadowing properties of discrete dynamical systems generated by diffeomorphisms of closed smooth manifolds. We show that the $C^1$-interior of the set of all diffeomorphisms having so-called inverse periodic shadowing property coincides with the set of $Ω$-stable diffeomorphisms. The equivalence of Lipschitz inverse periodic shadowing property and hyperbolicity of the closure of all periodic points is proved. Besides, we prove that the set of all diffeomorphisms that have Lipschitz inverse periodic shadowing property and whose periodic points are dense in the nonwa…

Abundance of fast growth of the number of periodic points in 2-dimensional area-preserving dynamics

We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there exists an open subset of the set of smooth area-preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points.

Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology

One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in the analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be approximated by ones with zero Lyapunov exponent in the space of ${\cal C}^ l$ ($1 \le l\le \infty$) smooth quasi-periodic cocycles. As a consequence, the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not ${\cal C}^ l$ open.

可解群的特征标、Hilbert-Schmidt 稳定性与稠密周期测度

Characters of solvable groups, Hilbert-Schmidt stability and dense periodic measures

本文利用亚阿贝尔群与多循环群的特征标理论及紧阿贝尔群自同构动力学，证明多类可解群具有 Hilbert-Schmidt 稳定性。

星流的周期轨道至多呈指数增长

At most exponential growth of periodic orbits for star flows

该研究证明闭流形上存在C1开且稠密的星流子集，其周期轨道至多指数增长，且方法可应用于含Lorenz多项式方程的截面双曲吸引子。

单频解析拟周期（埃尔米特）辛上循环的双曲性

Hyperbolicity for one-frequency analytic quasi-periodic (Hermitian)-symplectic cocycles

本研究针对实解析单频拟周期Sp(4,R)-上循环类，证明存在以李雅普诺夫指数互异或加速度非零为特征的开稠子集，部分解答了A. Avila提出的公开问题。

Remarks on definitions of periodic points for nonautonomous dynamical system

Let $(X,f_{1,\infty})$ be a nonautonomous dynamical system. In this paper we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new, very natural, definition of asymptotic periodicity. Moreover, this definition is resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We a…

Multi-chaos from Quasiperiodicity

One of the common characteristics of chaotic maps or flows in high dimensions is "unstable dimensional variability", in which there are periodic points whose unstable manifolds have different dimensions. In this paper, in trying to characterize such systems we define a property called "multi-chaos". A set $X$ is multi-chaotic if $X$ has a dense trajectory and for at least 2 values of $k$, the $k$-dimensionally unstable periodic points are dense in $X$. All proofs that such a behavior holds have been based on hyperbolicity in the sense that (i) there is a chaotic set $X$ with a dense trajectory…

Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms

We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic poin…

Periodic points and shadowing property for generic Lebesgue measure preserving interval maps

We show that for the generic continuous maps of the interval and circle which preserve the Lebesgue measure it holds for each k $\ge$ 1 that the set of periodic points of period k is a Cantor set of Hausdorff dimension zero and of upper box dimension one. Furthermore, building on this result, we show that there is a dense collection of transitive Lebesgue measure preserving interval map whose periodic points have full Lebesgue measure and whose periodic points of period k have positive measure for each k $\ge$ 1. Finally, we show that the generic continuous maps of the interval which preserve …

Periodic Points on the 2-sphere

For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the 2-sphere, we show that $f$ has $2^{n}$ periodic points.

论非双曲遍历测度作为周期测度极限的存在性

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

本文利用跟踪引理在无扰动条件下统一重构 [GIKN] 与 [BBD1] 的构造，证明鲁棒传递非双曲及鲁棒环微分同胚的开稠子集上存在可由双曲周期测度逼近的非双曲遍历测度。

Dense chaos for continuous interval maps

A continuous map $f$ from a compact interval $I$ into itself is densely (resp. generically) chaotic if the set of points $(x,y)$ such that $\limsup_{n\to+\infty}|f^n(x)-f^n(y)|>0$ and $\liminf_{n\to+\infty} |f^n(x)-f^n(y)|=0$ is dense (resp. residual) in $I\times I$. We prove that if the interval map $f$ is densely but not generically chaotic then there is a descending sequence of invariant intervals, each of which containing a horseshoe for $f^2$. It implies that every densely chaotic interval map is of type at most $6$ for Sharkovsky's order (that is, there exists a periodic point of period …

Periodic points for amenable group actions on uniquely arcwise connected continua

We show that if $G$ is a countable amenable group acting on a uniquely arcwise connected continuum $X$, then $G$ has either a fixed point or a 2-periodic point in $X$.

Periodic Points of Ward-Veech Surfaces

For a non-arithmetic Veech surface, it is known that the set points having finite orbit under the Veech group, called the set of periodic points, is finite. However, few examples of these periodic point sets have been computed. In what follows, we compute new examples of periodic point sets for a family of non-hyperelliptic Veech surfaces with unbounded complexity and consider applications to classifying the holomorphic sections of certain surface bundles as well as applications to the study of infinitely generated Veech groups.

The periodic points of renormalization

It will be shown that the renormalization operator, acting on the space of smooth unimodal maps with critical exponent greater than 1, has periodic points of any combinatorial type.

Statistical properties of periodic points for infinitely renormalizable unimodal maps

For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $φ$ which is a continuous or a geometric potential $φ=-β\log|f'|$ ($β\in\mathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $φ$. In particular, it follows that all periodic points are equidistributed with respect to mea…

Conservative surface homeomorphisms with finitely many periodic points

The goal of the article is to characterize the conservative homeomorphisms of a closed orientable surface $S$ of genus $\geq 2$, that have finitely many periodic points. By conservative, we mean a map with no wandering point. As a particular case, when $S$ is furnished with a symplectic form, we characterize the symplectic diffeomorphisms of $S$ with finitely many periodic points.

Robust transitivity and density of periodic points of partially hyperbolic diffeomorphisms

We prove results related to robust transitivity and density of periodic points of Partially Hyperbolic Diffeomorphisms under conditions involving Accessibility and a property in the tangent bundle .

Existence of periodic points with real and simple spectrum for diffeomorphisms in any dimension

We prove that for any $C^r$ diffeomorphism, $f$, of a compact manifold of dimension $d>2$, $1\leq r\leq \infty$, admitting a transverse homoclinic intersection, we can find a $C^1$-open neighborhood of $f$ containing a $C^1$-open and $C^r$-dense set of $C^r$ diffeomorphisms which have a periodic point with real and simple spectrum. We use this result to prove that $C^r$-generically among $C^r$ diffeomorphisms with horseshoes, we have density of periodic points with real and simple spectrum inside the horseshoe. As a corollary, we obtain that generically in the $C^1$-topology the unique obstruc…

Recurrent and periodic points in dendritic Julia sets

We relate periodic and recurrent points in dendritic Julia sets. This generalizes well-known results for interval dynamics.

Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $ϕ: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $μ_{\max}$ supported on a periodic orbit that maximizes the integral of $ϕ$ among all $f$ invariant Borel probability measures.

$C1$-Genericity of Symplectic Diffeomorphisms and Lower Bounds for Topological Entropy

There is a $C^1$-residual (Baire second class) subset $\mathcal{R}$ of symplectic diffeomorphisms on $2d$-dimensional manifold, $d\geq 1$, such that for every non-Anosov $f$ in $\mathcal{R}$ its topological entropy is lower bounded by the supremum of the Lyapunov exponents of their hyperbolic periodic points in the \emph{unbreakable central subbundle} (i.e., central direction with no dominated splitting) of $f$. The previous result deals with the fact that for $f$ in a residual set $\tilde{\mathcal{R}}$ of symplectic diffeomorphisms (containing $\mathcal{R}$) satisfies a trichotomy: or $f$ is …

Periodic points of Ruelle-expanding maps

We prove that, for a Ruelle-expanding map, the zeta function is rational and the topological entropy is equal to the exponential growth rate of the periodic points.

On the indices of periodic points in $C^1$-generic wild homoclinic classes in dimension three

We study the dynamics of homoclinic classes on three dimensional manifolds under the robust absence of dominated splittings. We prove that if such a homoclinic class contains a volume-expanding periodic point, then, $C^1$-generically, it contains a hyperbolic periodic point whose index (dimension of the unstable manifold) is equal to two.

无穷远处等于非退化线性映射的哈密顿微分同胚的周期点

Periodic Points of Hamiltonian Diffeomorphisms Equal to Nondegenerate Linear Maps at Infinity

本文研究了辛欧氏空间上在无穷远处等于非退化线性映射的哈密顿微分同胚，证明了在存在满足扭曲条件的孤立同调非平凡不动点时，必定存在无穷多个具有任意大素数周期的简单周期点。

Flexible periodic points

We define the notion of $\varepsilon$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits $\varepsilon$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that $\varepsilon$-perturbation to an $\varepsilon$-flexible point allows to change it in a stable index one periodic point whose (one dimensional) stable manifold is an arbitrarily chosen $C^1$ -curve. We also show that the existence of flexible point is a general phenomenon among systems with a robustly non-hyperbolic …

关于具有余维1扩张吸引子的微分同胚的孤立周期点

On isolated periodic points of diffeomorphisms with expanding attractors of codimension 1

研究$Ω$-稳定3-微分同胚中孤立周期点与余维1扩张吸引子的关系，利用吸引子结构估计孤立周期点的最小数量。

On the Number of Periodic Points of Quadratic Dynamical Systems Modulo a Prime

In 2004 Vasiga and Shallit studied the number of periodic points of two particular discrete quadratic maps modulo prime numbers. They found the asymptotic behaviour of the sum of the number of periodic points for all primes less than some bound, assuming the Extended Riemann Hypothesis. Later that same year Chou and Shparlinski proved this asymptotic result without assuming any unproven hypotheses. Inspired by this we perform experiments and find a striking pattern in the behaviour of the sum of the number of periodic points for quadratic maps other than the two particular ones studied previou…

Quasi-periodic solutions for differential equations with an elliptic-type degenerate equilibrium point under small perturbations

This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity.

Markov-Dyck shifts, neutral periodic points and topological conjugacy

We study the neutral periodic points of the Markov-Dyck shifts of finite strongly connected directed graphs. Under certain hypothesis on the structure of the graphs we show, that the topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs.

超越映射的单连通Fatou分支的周期边界点

Periodic boundary points for simply connected Fatou components of transcendental maps

该研究针对超越映射的单连通吸引盆、抛物盆或双抛物Baker域，在后奇异集的特定假设下证明其边界上周期点稠密，推广了有理映射的相关结果，还给出了具独立价值的内函数单位圆周附近畸变结果。