A. Baraviera, Daniel Gonçalves, Danilo Royer, Ruy Exel, Fagner B. Rodrigues
In this paper we introduce the definition of entropy for a partial $\mathbb{Z}$-action. We show that the definition of partial entropy is an extension of the definition of topological entropy for a $\mathbb Z$-action. We also prove that the partial topological entropy is concentrated on the non-wandering set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Entropy Formula for Random $\mathbb{Z}^k$-actions
本文考虑了随机$\mathbb{Z}^k$作用的熵，包括测度熵和拓扑熵，这些作用是由$\mathbb{Z}^k$作用的生成元的随机合成所产生的.应用Pesin的交换同构理论，通过生成元的李雅普诺夫谱，得到了$C^{2}$随机$\mathbb{Z}^k$作用的测度论熵公式.对于由Lipschitz映射、有限图上的连续映射和$C^{1}$扩张映射等更一般的映射生成的随机$\mathbb{Z}^k$（或$\mathbb{Z}_+^k$）作用，也得到了拓扑熵的一些公式和界.此外，作为应用，我们给出了某些$C^{2}$ $\mathbb{Z}^k$作用的Friedland熵公式.
A Study of Directional Entropy Arising from \(\mathbb{Z} \times \mathbb{Z}_+\) Semigroup Actions
本文针对一维 LCA 与移位变换生成的半群作用，分别在 Milnor 几何框架和 Kolmogorov–Sinai 形式体系下系统考察拓扑与测度论方向熵。
No smooth Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with positive entropy
对$\mathbb{C}^2$的任意正熵多项式同态$f$，其Julia集及其逆$f^{-1}$的Julia集都不是$C^1$光滑的有边界流形.
Entropy of a unitary operator on $l^2(\mathbb{N}_0)$
该研究推广已有熵理论，计算更广泛有限维算子的熵，并用其逼近及具体求解 $l^2(\mathbb{N}_0)$ 上酉算子等若干算子的熵。
$k$-type entropy of $\mathbb{Z}^d$ actions
本文提出$k$-型熵的概念，研究其性质并计算了二维环面上$\mathbb{Z}^2$作用的熵。
Bounding entropy for one-parameter diagonal flows on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ using linear functionals
本文提出一种利用尖点区域的熵贡献界定 $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ 上单参数对角不变测度熵的方法，该界依赖于 Cartan 群李代数上的辅助线性泛函，后续将证明其可优化且在许多情况下是紧的。
Polynomial entropy of induced homeomorphisms on $C_k(\mathbb{S}^1)$ and $C_k([0,1])$
本文研究了圆同胚诱导映射在$C(\mathbb{S}^1)$上的多项式熵计算问题。
In this paper, unstable metric entropy, unstable topological entropy and unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. A version of Shannon-McMillan-Breiman Theorem is established, and a variational principle is formulated, which gives a relationship between unstable metric entropy and unstable pressure (unstable topological entropy). As an application of the variational principle, some results on the $u$-equilibrium states are given.
In this article we study the regularity of the topological and metric entropy of partially hyperbolic flows with two-dimensional center direction. We show that the topological entropy is upper semicontinuous with respect to the flow, and we give an example where the lower semicontinuity fails. We also show that if such a flow has no fixed points, then it is entropy expansive, and consequently the metric entropy function is upper semicontinuous, there exist equilibrium states (and measures of maximal entropy), and principal symbolic extensions.
Completely positive entropy actions of sofic groups with $\mathbb{Z}$ in their center
设$Γ$是一个sofic群，其中心有一个$\mathbb{Z}$的副本。我们构造了一个不可数的两两非同构的测度保持$Γ$行动与完全正熵，其中没有一个因素的伯努利位移。与Bernoulli位移之间的同构关系相反，我们的构造表明完全正熵$Γ$作用之间的同构关系是不光滑的.
We consider the category of partially observable dynamical systems, to which the entropy theory of dynamical systems extends functorially. This leads us to introduce quotient-topological entropy. We discuss the structure that emerges. We show how quotient entropy can be explicitly computed by symbolic coding. To do so, we make use of the relationship between the category of dynamical systems and the category of graphs, a connection mediated by Markov partitions and topological Markov chains.
For a circle map $f\colon\mathbb{S}\to\mathbb{S}$ with zero topological entropy, we show that a non-diagonal pair $\langle x,y\rangle\in \mathbb{S}\times \mathbb{S}$ is non-separable if and only if it is an IN-pair if and only if it is an IT-pair. We also show that if a circle map is topological null then the maximal pattern entropy of every open cover is of polynomial order.
Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a compact Riemannian manifold $M$, in this paper we define the concept of unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily compact and we extend a theorem of R. Bowen proving that, for an ergodic $f$-invariant measure $μ$, the unstable measure theoretical entropy of $f$ is upper bounded by the unstable topological entropy of $f$ on any set of full $μ$-measure. We also define a notion of unstable topological entropy of $f$ using a Hausdorff dimension like characterization.
Upper bounds for the entropy in the cusp for one-parameter diagonal flows on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$
本文研究了 $SL_d(\mathbb{R})/SL_d(\mathbb{Z})$ 上单参数对角流的尖点熵问题，通过在 Cartan 群李代数上选取辅助线性泛函，给出了整体及不同抛物子群对应尖点区域的显式熵上界，并在后续工作中证明其紧性。
On Directional Entropy of a $\mathbb{Z}^{2}$-Action
考虑$\mathbb{Z}^{2}$ -作用$Φ$的元胞自动机（CA）在所有值在有限集合$\mathbb{Z}_{r}$，$r \geq 2$中的双无限序列空间上，由元胞自动机$T_{F[-k, k]}$和加法自动机规则$F(x_{n-k},...,x_{n+k})=\sum\limits_{i=-k}^{k}a_{i}x_{n+i}(mod r)$确定.研究了$ω_{0}$方向上单位长度的测度理论方向熵的概念。证明了$h_μ(T_{F[-k,k]}^{u})=uh_μ(T_{F[-k,k]})$，$h_μ(Φ^{u})=uh_μ(Φ)$和$h_{\vec{v}}(Φ^{u})=uh_{\vec{v}}(Φ)$对于$\vec{v} \in \mathbb{Z}^{2}$，其中$h$是测度论熵.
In this work we study the dynamical behavior Tonelli Lagrangian systems defined on the tangent bundle of the torus $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. We prove that the Lagrangian flow restricted to a high energy level $ E_L^{-1}(c)$ (i.e $ c> c_0(L)$) has positive topological entropy if the flow satisfies the Kupka-Smale propriety in $ E_L^{-1}(c)$ (i.e, all closed orbit with energy $c$ are hyperbolic or elliptic and all heteroclinic intersections are transverse on $E_L^{-1}(c)$). The proof requires the use of well-known results in Aubry-Mather's Theory.
In this paper, we first concentrate on the possible values and dense property of entropies for isotropic and anisotropic axial products of subshifts of finite type (SFTs) on $\mathbb{N}^d$ and $d$-tree $\mathcal{T}_d$. We prove that the entropies of isotropic and anisotropic axial products of SFTs on $\mathbb{N}^d$ are dense in $[0,\infty)$, and the same result also holds for anisotropic axial products of SFTs on $\mathcal{T}_d$. However, the result is no longer true for isotropic axial products of SFTs on $\mathcal{T}_d$. Next, motivated by the work of Johnson, Kass and Madden [16], and Schra…
Entropy of Lyapunov maximizing measures of $SL(2,\mathbb{R})$ typical cocycles
针对满足挤压、扭转及Mather集上映射一一条件的一步$SL(2,\mathbb{R})$典型上循环，论文证明其Lyapunov最大化测度熵为零。
Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms
研究了部分双曲型方程组不稳定部分引起的熵。本文定义了不稳定度量熵和不稳定拓扑熵，建立了部分双曲型同态的一个变分原理，即不稳定拓扑熵是不稳定度量熵在所有不变测度上的上确界.不变测度的不稳定度量熵被定义为沿非稳定流形的条件熵，它与Ledrappier-Young给出的度量熵相同，尽管我们没有使用递增分割.通过不稳定叶理上的分离集、生成集和开覆盖等价地定义了不稳定拓扑熵，它与不稳定叶理上的不稳定体积增长相一致.我们还得到了不稳定度量熵的一些性质，如仿射性、上半连续性和Shannon-McMillan-Breiman定理.
Dominated Splitting, Partial Hyperbolicity and Positive Entropy
设$f:M\rightarrow M$是无边界紧致黎曼流形$M$上的一个具有控制分裂的$C^1$单同态.给出并证明了$f$拓扑熵为正的几个充分条件。条件处理的动力学行为（不一定不变）勒贝格测度。特别地，如果勒贝格测度是$δ$ -递归的，则$f$的熵是正的。我们给出反例说明这些充分条件是不必要的。最后，在部分双曲同态的情形下，我们给出了熵与不稳定和稳定子丛维数之间的正下界。
In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb{Z}^q$-systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, direcitonal discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb{Z}^q$-system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb{Z}^q$-system has directional discrete spectrum along $q$ linearly independent directions if and only if it has discrete spectrum.
On an entropy of $Z_+^k$-actions
本文给出了S.研究了Friedland \cite{Friedland}。与传统的定义不同，它可以为生成元具有有限（甚至为零）熵作为单个变换的动作取非零值。研究了扩张自同态生成的圆上的$\mathbb{Z}_+^k$ -作用的一些基本性质，并给出了它的值.此外，通过原像熵，这是依赖于系统的“逆轨道”结构的类熵不变量，通过扩展自同态生成的环面上的$\mathbb{Z}_+^k$ -作用的熵的上界得到。
Entropy stability for diffeomorphisms isotopic to Anosov on $\mathbb{T}^d$
本文研究d维环面上与线性Anosov自同构同痕的部分双曲微分同胚的拓扑熵，证明一类DA映射的拓扑熵与线性映射一致，得到低振荡势的平衡态唯一性等结果，还构造出拓扑熵严格大于线性部分的动态相容微分同胚。
Mean dimension of $\mathbb{Z}^k$-actions
平均维数是动力系统的一个拓扑不变量，它对具有无穷维和无穷熵的系统是有意义的。给定紧致度量空间$X$上的一个$\mathbb{Z}^k$ -作用，我们研究了以下三个与平均维数密切相关的问题. (1)什么时候$X$同构于有限熵系统的逆极限？(2)假设拓扑熵$h_{\mathrm{top}}(X)$是无限的。如果只考虑$X$到给定的精度水平，那么可以检测到多少拓扑熵？随着分辨率越来越精细，熵的增长速度有多快？(3)我们什么时候可以将$X$嵌入到无限维立方体$([0,1]^D)^{\mathbb{Z}^k}$上的$\mathbb{Z}^k$ -移位中？在[Lindenstrauss，Mean dimension，small entropy factors and an embedding theorem，Inst. Hautes Etudes Sci. Publ. Math. \textbf{89}（1999）227-262]，但是推广到$\mathbb{Z}^k$仍然是一个开放的问题。当$X$具有标记性质时，特别是当$X$具有完全非周期的最小因子时，我们完全解决了（1）和（2）的自然解释，并给出了（3）的合理满意的答案。一个关键因素是一种新的方法，不断分割成良好的每一个轨道部分。
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…
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: $$h_{\rm{top}}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int\max_{V\subset T_{x}M}|\det Df_{x}^{n}|_{V}|\,d x.$$
In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in \cite{Bo, PS2007}. Our results give new insights to the multi…
Continuity properties of partial entropy
本文研究了$C^{1+α}$微分同胚中偏熵的上半连续性，证明了当Lyapunov指数之和连续时该性质成立，扩展了Newhouse关于光滑微分同胚的结果。
Directional uniformities, periodic points, and entropy
由$d\ge2$交换同胚（拓扑$\mathbb{Z}^d$ -作用）生成的动力系统在它们内部包含许多尺度上的结构，特别是包含$1\le k\le d$的$\mathbb{Z}^k$的许多作用。同胚的熟悉的动力学不变量，如熵和周期点数据，变得更加复杂，并允许多个定义。我们简要地调查了一些这些和其他相关的不变量设置的代数$\mathbb{Z}^d$ -行动，显示如何，即使在设置的自然熵作为$\mathbb{Z}^d$ -行动消失，一个强大的理论方向熵和周期点可以建立。一个基本的主题是一致性的动力学不变量的方向变化，以及这个理论和问题之间的联系，在数论，我们探讨这几个不变量。我们还强调弗里德的平均熵的概念和它的连接均匀的增长特性，并证明了这种熵和周期点增长在这种设置之间的新关系。
We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not a conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(\mathcal{T}_X) \geq h(X)$, where $\mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two…
Boundary complexity and surface entropy of 2-multiplicative integer systems on $\mathbb{N}^d$
本文针对 $\mathbb{N}^d$ 上的 2-乘性整数系统，引入边界复杂度概念，证明特定区间内所有值均可实现，并给出曲面熵的严格公式，从而提供计算拓扑熵的有效方法并揭示其与有限型子移位的内在差异。
Smooth ergodic theory of $\mathbb{Z}^d$-actions
在本文的第一部分中，我们制定了一个一般的设置中，研究可微的$\mathbb{Z}^d$ -行动保持一个Borel概率测度的遍历论。这个框架包括了紧致流形的$C^{1+\text{Hölder}}$仿射的作用。我们构造中间和粗非稳定流形的行动，并建立控制他们的局部几何。在第二部分中，我们考虑的熵，李雅普诺夫指数，和几何条件的措施秩1系统的Ledrappier-杨熵公式的一些推广之间的关系。在第三部分中，我们证明了对于保Borel概率测度的$\mathbb{Z}^d$的光滑作用，熵沿粗糙非稳定流形满足一定的“乘积结构”。此外，给定两个光滑的$\mathbb{Z}^d$ -行动-其中之一是一个可测量的因素，其他-我们表明，所有粗糙的李雅普诺夫指数有助于熵的因素系统是粗糙的李雅普诺夫指数的总系统。作为结果，我们导出了粗糙非稳定流形的熵的Abramov-Rohlin公式。
Toeplitz Subshifts with Trivial Centralizers and Positive Entropy
给定一个动力系统$(X,G)$，中心化子$C(G)$表示$X$的所有同胚的群，这些同胚与$G$的作用可交换。这个群有时被称为动力系统$(X,G)$的自同构群。本文通过证明一类具有平凡中心化子的$\mathbb Z^d$-Toeplitz系统，将Bulatek和Kwiatkowski（1992）的构造推广到$\mathbb Z^d$-Toeplitz系统.我们证明了这类具有平凡中心化器的$\mathbb Z^d$ -Toeplitz包含具有正拓扑熵的系统。
On entropy of $\mathbb{P}$-twists
本文通过计算范畴熵函数，证明光滑射影簇上 $\mathbb{P}$-对象诱导的 $\mathbb{P}$-twist 不共轭于标准自等价，并在附加条件下确定其与 spherical twist 的范畴多项式熵。
In this paper we provide a concrete construction of Furstenberg entropy values of $τ$-boundaries of the group $\mathbb{Z}[\frac{1}{p_1},\ldots,\frac{1}{p_{l}}]\rtimes \{p_1^{n_1}\cdots p_{l}^{n_{l}} \, : \, n_i\in\mathbb{Z}\}$ by choosing an appropriate random walk $τ$. We show that the boundary entropy spectrum can be realized as the subsum-set for any given finite sequence of positive numbers.
Shannon entropy for stationary processes and dynamical systems
本文研究了由$\mathbb Z$或$\mathbb Z^n$表示的平稳遍历过程，其有限维边缘具有关于勒贝格测度绝对连续的律.我们定义了一个熵理论，这些连续的过程，证明了类似的香农Breiman Mac Millan定理和研究更精确的高斯过程的特定例子。
The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal and bimodal regions. Utilizing the theory of polynomial-like mappings, we prove that the level sets of the real entropy function $h_\mathbb{R}$ are connected in the $(-+-)$-bimodal region and a portion of the unimodal region in $\mathcal{M}_2(\mathbb{R})$. Based on the numerical evidence, we conjecture that the monotonicity holds throughout th…
We show that the automorphism group of every zero entropy infinite shift admits a "drift" homomorphism to $(\mathbb{R},+)$ that maps the shift map to 1. This homomorphism arises as the expectation, under an invariant measure, of a cocycle defined on a space of asymptotic pairs.
Ergodic measures of intermediate entropies for $\mathbb{Z}^{d}$-action
本文研究满足近似$\mathbb{Z}^{d}$或$\mathbb{Z}_+^{d}$-积性质且渐近熵扩张的动力系统，证明了其不变测度空间的结构，并证明了具有任意给定中间熵的遍历测度在某些自然子空间中是稠密的，从而验证了Katok关于此类系统存在任意中间熵遍历测度的猜想。
A note on two types of Lyapunov exponents and entropies for $\mathbb{Z}^k$-actions
本文考虑光滑$\mathbb{Z}^k$作用的两类李雅普诺夫指数：随机李雅普诺夫指数和方向李雅普诺夫指数，以及相应的熵：随机熵和方向熵.通过生成元的李雅普诺夫指数，研究了这些量之间的密切关系，并给出了它们的计算公式。
In this short paper, we compute the directional sequence entropy for of $\mathbb{Z}_+\times \mathbb{Z}$-actions generated by cellular automata and the shift map. Meanwhile, we study the directional dynamics of this system. As a corollary, we prove that there exists a sequence such that for any direction, some of the systems above have positive directional sequence entropy. Moreover, with help of mean ergodic theory for directional weak mixing systems, we obtain a result of number theory about combinatorial numbers.
Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of $\mathbb{R}^k$
本文证明维数 $2k+1$ 流形上 $\mathbb{R}^k$ 光滑极大秩作用在所有非平凡时间映射均对某遍历测度有正熵时具有算术性，即测度同构于仿射 Cartan $\mathbb{Z}^k$ 作用悬置的常数时间变换。
We prove that if two free p.m.p. $\mathbb{Z}$-actions are Shannon orbit equivalent then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein-Weiss and the entropy invariance results of Austin and Kerr-Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also …
In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $α\in [g,G]$, where $g=\tfrac{1}{2}(\sqrt{5}-1)$ and $G=g+1=1/g$ are the two golden mean numbers is introduced. In this article, by using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [BM] are obtained, and it is shown that for each $α,α^*\in [g,G]$ the natural extensions from [BM] are …
A link between Topological Entropy and Lyapunov Exponents
我们证明了一般的非部分双曲辛同态$C^1-$ $f$的拓扑熵等于其双曲周期点的正李雅普诺夫指数之和的上确界.此外，我们还证明了$f$的拓扑熵近似于$f$的拓扑熵限制在基本双曲集上.特别地，拓扑熵映射在远离部分双曲性的辛同态的$C^1-$脱殊集中是下连续的。
We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for $\text{SL}_2(\mathbb{R})$ to disintegrations of stationary measures for $\text{GL}(\mathbb{R}^d)$ onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace. The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simpl…
The Topological Directional Entropy of Z^2-actions Generated by Linear Cellular Automata
本文利用生成的加法元胞自动机研究了$\mathbb{Z}^2$ -作用的拓扑熵和度量方向熵（CA），由局部规则$f[l, r]$、$l, r\in \mathbb{Z}$、$l\leq r$定义，即由环$\mathbb{Z}_m (m \geq 2)$上的$T_{f[l, r]}(x) =(y_n)_ {-\infty}^{\infty}$、$y_{n} = f(x_{n+l}, ..., x_{n+r}) = \sum_{i=l}^rλ_{i}x_{i+n}(mod m)$、$x=(x_n)_ {n=-\infty}^{\infty}\in \mathbb{Z}^\mathbb{Z}_{m}$和$f: \mathbb{Z}_{m}^{r-l+1}\to \mathbb{Z}_{m}$给出的映射$T_{f[l, r]}: \mathbb{Z}^\mathbb{Z}_{m} \to \mathbb{Z}^\mathbb{Z}_{m}$，以及作用在紧致度量空间$\mathbb{Z}^\mathbb{Z}_{m}$上的移位映射，其中$m$ $(m \geq2)$是正整数。我们的主要目的是给出一个算法来计算由加性CA和移位映射生成的$\mathbb{Z}^2$ -作用的拓扑方向熵。因此，我们要求给出一个封闭的公式的拓扑方向熵的$\mathbb{Z}^2$ -行动产生的对$(T_{f[l, …
We construct Z^2-SFTs at every computable level of the hierarchy of topological completely positive entropy (TCPE), answering Barbieri and García-Ramos, who asked if there was one at level 3. Furthermore, we show the property of TCPE in Z^2-SFTs is coanalytic complete. Thus there is no simpler description of TCPE in Z^2-SFTs than in the general case.
Topological Entropy and Partially Hyperbolic Diffeomorphisms
我们考虑紧致流形上的部分双曲型同胚，其中不稳定和稳定的叶理稳定地携带一些唯一的非平凡同调。证明了如下两个结果：若中心叶理是一维的，则拓扑熵是局部常数;若中心叶理是二维的，则拓扑熵在所有$C^\8$拓扑同态的集合上是连续的.证明中使用了我们引入的一个拓扑不变量、Yomdin关于上半连续性的定理、Katok关于二维系统下半连续性的定理以及我们证明的部分双曲型同态的一个精化Pesin-Ruelle不等式。
Projectional entropy and the electrical wire shift
本文给出了有限型$W^{\text{el}}$的一个可扩展的块胶合$\mathbb Z^3$移位，其中对于二维子格$L:=\mathbb Z \vec{e}_1+\mathbb Z\vec{e}_2\subsetneq\mathbb Z^3$，其拓扑熵等于$L$ -投影熵，但$W^{\text{el}}$不是$W^{\text{el}}_L$的完全$\mathbb Z$扩展.特别地，这个例子表明[3]的定理4.1不能推广到$r>1$的$r$维子格$L$.然而，我们能够在与[3]相同的混合假设下，重新证明和推广关于一般（非SFT）$\mathbb Z^d$移位的一维子格的结果，并且通过提出更强的混合条件，我们也得到了高维子格的相应陈述。
Entropy on modules over the group ring of a sofic group
对于给定的有限域${\mathbb F}$和sofic群$Γ$，部分推广了群环${\mathbb F} Γ$上模的Peters公式.讨论了熵值与零因子猜想的关系。
Topological Entropy Conjecture
1974年，M. Shub提出了拓扑熵猜想，即不等式$\logρ\leq ent(f)$是否成立，其中$f$是紧致流形$M$上的连续自映射，$ent(f)$是$f$的拓扑熵，$ρ$是$f$在同调群$H_{*}(M;\mathbb{Z})=\bigoplus\limits_{i=0}^n{H_{i}(M;\mathbb{Z})}$上诱导的线性变换$f_*$的最大绝对特征值. 1986年，A. B. Katok给出了一个反例，使得不等式$\logρ\leq ent(f)$是无效的.本文在紧致Hausdorff空间$X$上定义了$f$ -Čech同调群$\check{H}_{i}(X,f;\mathbb{Z})$和拓扑纤维熵$ent(f_L)$，其中$n=n(J)$使得$\check{H}_*(X;\mathbb{Z})$存在，其中$f\in C^0(X)$和$J$是所有覆盖的集合.然后证明了$\logρ\leq ent(f_L)$是有效的。
On entropy, regularity and rigidity for convex representations of hyperbolic manifolds
给出了凸余紧群$Γ$的一个凸表示$ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$，给出了$\mathbb{H}^k$的量$αh_ρ,$的上界，其中$h_ρ$是$ρ$的熵，$α$是等变映射$\partialΓ\to\mathbb{P}(\mathbb{R}^d).$的Hölder指数.然后，我们研究希钦表示和证明，如果$ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$是在希钦组件，然后$αh_ρ\leq 2/(d-1)$（其中$α$是Hölder指数的地图$ζ:\partial\mathbb{H}^2\to\mathscr{F}$）与平等当且仅当$ρ$是Fuchsian。
Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$
本文证明了一个凸域上的Blaschke距离与Hilbert距离之间的尖锐不等式：对任意两点$x$和$y$，\[d^B(x,y) < d^H(x,y) +1.\]我们得到两个有趣的结果：第一个是Hilbert几何的体积熵刚性：对$\mathbb{R}\mathbf{P}^n$的任意凸域，半径为$R$的球的体积至多象$e^{(n-1)R}$那样增长.第二个结果是如下事实：对于曲面群在$\mathrm{PSL}(3,\mathbb{R})$中的任何希钦表示，在$\mathrm{PSL}(2,\mathbb{R})$中存在Fuchsian表示$j$，使得$j$的长度谱均匀小于$ρ$的长度谱。
Partial hyperbolicity and foliations in $\mathbb{T}^3$
证明了$\mathbb{T}^3$与Anosov同构的部分双曲双同态空间的动力学相干性是一个开的和闭的性质。证明了$\mathbb{T}^3$的强部分双曲同态要么是动态相干的，要么有一个不变的收缩或排斥的二维环面.为此，我们开发了一些一般性的结果余维一叶理，这可能是独立的利益。
We study diagonal actions $φ:\mathbb{F}_2\curvearrowright\partial\mathbb{F}_2\times K$ on the Cantor set which are given by $φ_a=\partial_a\timesα,φ_b=\partial_b\timesβ$. Under some restrictions on $α,β$ we compute $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)$. As an application in the case of $α$ is Denjoy homeomorphism of the Cantor set and $β=id$ we will show that $C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2$ is Kirchberg algebra with $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)=(\mathbb{Z}^\infty,\mathbb{Z}^\infty)$. Also we will check that $C^*$-crosse…
Fuglede-Kadison determinants and entropy for actions of discrete amenable groups
在1990年，Lind，Schmidt和Ward给出了一个关于某些$\mathbb{Z}^n$ -动力系统的熵的公式，该动力系统附着于Laurent多项式$P$，根据$P$的（对数）Mahler测度。我们扩展的情况下，他们的结果的非交换设置$\mathbb{Z}^n$得到适当的离散顺从群取代。推广马勒测度，Fuglede-Kadison行列式群冯诺依曼代数理论出现在熵公式。
Large entropy measures for endomorphisms of CP(k)
设$f$是$\mathbb{C}\mathbb{P}^k$的全纯自同态.利用编码技术构造了一类遍历测度作为点原像上非一致概率测度的极限。我们发现，他们有大的度量熵，接近$\log d^k$。我们建立了它们的强随机性质，并证明了它们的李雅普诺夫指数的正性。由于它们具有大的熵，所以这些度量在$f$的最大熵度量的支持下得到支持。它们特别提供了朱利亚集的豪斯多夫维数的下界。
We prove the following statement: Let $X=\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})$, and consider the standard action of the diagonal group $A<\text{SL}_n(\mathbb{R})$ on it. Let $μ$ be an $A$-invariant probability measure on $X$, which is a limit $$ μ=λ\lim_i|ϕ_i|^2dx, $$ where $ϕ_i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$, and $λ>0$ is some positive constant. Then any regular element $a\in A$ acts on $μ$ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over $\m…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。