Topological entropy of a magnetic flow and the growth of the number of trajectories

Closed billiard trajectories with prescribed bounces

We give the asymptotic growth of the number of primitive periodic trajectories of a two dimensional dispersive billiard, when we prescribe their number of bounces on one of the obstacles.

Uniform Periodic Point Growth in Entropy Rank One

We show that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions.

关于Floer条形码的增长

On the growth of the Floer barcode

该研究引入序列条形码熵分析紧支撑哈密顿微分同胚迭代的Floer条形码增长，证明二维下其等于拓扑熵与普通条形码熵，同时探究了迭代下γ-范数的分离性质及相关映射的条形码熵特性。

增长阶与广义熵

Orders of Growth and Generalized Entropy

本文构造了完整的增长阶集合，并定义广义熵，以研究超越指数增长的轨道分离，尤其适用于零熵系统，并揭示相关空间比预期更复杂。

Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms

We show that the metric entropy of a $C^1$ diffeomorphism with a dominated splitting and the dominating bundle uniformly expanding is bounded from above by the integrated volume growth of the dominating (expanding) bundle plus the maximal Lyapunov exponent from the dominated bundle multiplied by its dimension. We also discuss different types of volume growth that can be associated with an expanding foliation and relationships between them, specially when there exists a closed form non-degenerate on the foliation. Some consequences for partially hyperbolic diffeomorphisms are presented.

Ensemble-based Topological Entropy Calculation (E-tec)

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how trajectories "braid" around one another. In this spirit, we introduce the Ensemble-based Topological Entropy Calculation, or E-tec, a method to derive a lower-bound on topological entropy of two-dimensional systems by considering the evolution of a "rubber band" (piece-wise linear cu…

Entropy and periodic orbits for equivalent smooth flows

Given any $K>0$, we construct two equivalent $C^2$ flows, one of which has positive topological entropy larger than $K$ and admits zero as the exponential growth of periodic orbits, in contrast, the other has zero topological entropy and super-exponential growth of periodic orbits. Moreover we establish a $C^{\infty}$ flow on $\mathbb{S}^2$ with super-exponential growth of periodic orbits, which is also equivalent to another flow with zero exponential growth of periodic orbits. On the other hand, any two dimensional flow has only zero topological entropy.

链补空间中的拓扑熵与轨道增长

Topological entropy and orbit growth in link complements

本文研究闭三维流形上正熵流的周期轨道环绕结构，构造链补空间中同伦类唯一的周期轨道马蹄，并用于刻画 generic Reeb 流的拓扑熵与接触同调增长。

Non-existence of a universal zero entropy system via generic actions of almost complete growth

We prove that a generic p.m.p. action of a countable amenable group $G$ has scaling entropy that can not be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of $G$ for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. $G$-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that can not be bounded from below by a given sequence. We also show an example of an amenable group that has such lower bound for every free p.m.p. action.

度量熵与同宿增长速率

Metric entropy and homoclinic growth rate

本文针对紧致黎曼流形上保持遍历双曲测度的C^r微分同胚，证明同宿增长速率以度量熵为下界，推广了Mendoza的曲面结果至高维，并通过Newhouse域的例子检验了估计的尖锐性。

Entropy, volume growth and SRB measures for Banach space mappings

We consider $C^2$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volum…

Volume growth and topological entropy of certain partially hyperbolic systems

Let $f$ be a $C^{1}$ diffeomorphism on a compact manifold $M$ admitting a partially hyperbolic splitting $TM=E^{s}\oplus_{\prec} E^{1}\oplus_{\prec} E^{2}\cdots \oplus_{\prec}E^{l}\oplus_{\prec} E^{u}$ where $E^{s}$ is uniformly contracting, $E^{u}$ is uniformly expanding and $\dim E^{i}=1,\,1\leq i\leq l.$ We prove an entropy formula w.r.t. the volume growth rate of subspaces in the tangent bundle: $$h_{\rm{top}}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int\max_{V\subset T_{x}M}|\det Df_{x}^{n}|_{V}|\,d x.$$

Entropy conjugacy for Markov multi-maps of the interval

We consider a class $\mathcal{F}$ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $[0,1]^{\mathbb{Z}_+}$. For a multi-map in $\mathcal{F}$, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $\mathcal{F}$.

拓扑熵增长与混沌吸引子形状全局变化的机制

A mechanism for growth of topological entropy and global changes of the shape of chaotic attractors

本文研究了连续时间动力系统中导致拓扑熵无界增长的吸引子结构全局变化机制，以经典Roessler系统为例，通过计算机辅助证明确立了特定参数下混沌吸引子的存在及引发拓扑熵增长的分岔序列。

正则集上的熵与奇异流的周期轨道增长

Entropy on regular sets and periodic-orbit growth for singular flows

该研究针对度量空间上的流，证明紧度量空间上流的有限拓扑熵由正则集紧子集承载，建立Bowen–Walters不等式并将其应用于多类奇异流，扩展了相关文献的结论。

Entropy and growth rate of periodic points of algebraic Z^d-actions

Expansive algebraic Z^d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.

Topological entropy, homological growth and zeta functions on graphs

In connection with the Entropy Conjecture it is known that the topological entropy of a continuous graph map is bounded from below by the spectral radius of the induced map on the first homology group. We show that in the case of a piecewise monotone graph map, its topological entropy is equal precisely to the maximum of the mentioned spectral radius and the exponential growth rate of the number of periodic points of negative type. This nontrivially extends a result of Milnor and Thurston on piecewise monotone interval maps. For this purpose we generalize the concept of Milnor-Thurston zeta fu…

Spectrum and Entropy of C-systems. MIXMAX random number generator

The uniformly hyperbolic Anosov C-systems defined on a torus have very strong instability of their trajectories, as strong as it can be in principle. These systems have exponential instability of all their trajectories and as such have mixing of all orders, nonzero Kolmogorov entropy and a countable set of everywhere dense periodic trajectories. In this paper we are studying the properties of their spectrum and of the entropy. For a two-parameter family of C-system operators A(N,s), parametrised by the integers N and s, we found the universal limiting form of the spectrum, the dependence of en…

On the continuity of topological entropy of certain partially hyperbolic diffeomorphisms

In this paper, we consider certain partially hyperbolic diffeomorphisms with center of arbitrary dimension and obtain continuity properties of the topological entropy under $C^1$ perturbations. The systems considered have subexponential growth in the center direction and uniform exponential growth along the unstable foliation. Our result applies to partially hyperbolic diffeomorphisms which are Lyapunov stable in the center direction. It applies to another important class of systems which do have subexponential growth in the center direction, for which we develop a technique to use exponential…

Buildings, Extensions, and Volume Growth Entropy

Let F be a non-Archimedean local field and let E be a finite extension of F. Let G be a split semisimple F group. We discuss how to compare volumes on the Bruhat-Tits buildings B_E and B_F of G(E) and G(F) respectively.

对象的质量增长与范畴熵

Mass growth of objects and categorical entropy

本文研究三角范畴上自函子映射下对象质量增长率的基本性质，并阐明其与范畴熵的关系，证明在特定条件下二者一致。

Resonance and marginal instability of switching systems

We analyse the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Chitour, Mason, and Sigalotti (2012) stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. We provide a characterization of marginal instability under some mild assumptions on the sys- tem. These assumptions can be verified algorithmically and are believed to be generic. Finally…

秩一系统的序列熵

Sequence entropy of rank one systems

本文研究了秩一保测系统沿亚指数序列的序列熵，证明了在一定序列类中序列熵可以为无穷大，并展示了当塔高增长速率低于特定阈值时，亚指数序列的序列熵必然消失。

马勒度量与具有拉格朗日性质的动力系统及代数熵的增长

Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property

研究具有拉格朗日性质的动力系统和簇代数中马勒度量的增长率，比较其与其他增长度量，并提出马勒度量的增长率与代数熵一致的猜想。

Entrance Time and Rényi Entropy

For ergodic systems with generating partitions, the well known result of Ornstein and Weiss shows that the exponential growth rate of the recurrence time is almost surely equal to the metric entropy. Here we look at the exponential growth rate of entrance times, and show that it equals the entropy, where the convergence is in probability in the product measure. This is however under the assumptions that the limiting entrance times distribution exists almost surely. This condition looks natural in the light of an example by Shields in which the limsup in the exponential growth rate is infinite …

Degree-growth of monomial maps

For projectivizations of rational maps Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 imply that the algebraic entropy of a monomial map is always bounded above by its topo…

管状维数：叶向渐近局部乘积结构，以及熵与体积增长

Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth

本文引入管状维数并给出计算公式，利用该概念证明了微分同胚条件测度的渐近局部乘积结构，给出了条件熵间隙的体积增长界，并计算了特定映射的最大熵测度条件熵，为解决Binder-DeMarco猜想奠定基础。