Manfred Einsiedler, Thomas Ward
We show that many algebraic actions of higher-rank abelian groups on zero-dimensional groups are mutually disjoint. The proofs exploit differences in the entropy geometry arising from subdynamics and a form of Abramov--Rokhlin formula for half-space entropies.
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Quasi-similarity, entropy and disjointness of ergodic systems
回答 Vershik 问题，证明可数无穷群 Bernoulli 作用的拟相似性不保持熵，并推广 Pinsker 定理，证明零 $P$-熵与完全正 $P$-熵作用不相交。
On disjointness and $\mathscr{F}$-product recurrence with respect to zero entropy systems
本文研究零熵系统的不交性与 $\mathscr{F}$-乘积回复性，证明了相关等价关系，并构造例子分别肯定和否定地回答了两个开放问题。
Sarnak's Möbius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) μ(n)= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing that this $o(1)$ can go to zero arbitrarily slowly. In fact, our methods yield a more general result, where in lieu of $μ(n)$ one can put any bounded sequence such that the Cesàro mean of the corresponding sequence of absolute values does not tend to zero.
We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Folner sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. Meanwhile, we investigate the relations among them. Secondly, we introduce the notion of a relative dimension set. Moreover, using it, we discuss the disjointness between the relative entropy zero extensions which generalize…
The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This combinatorial approach provides us with a new insight to analyze the entropy zero systems. We also define the dimension set of a system to investigate the structure of the randomness of the factors of a system. The notion of a uniform dimension in the class of entropy zero systems is introduced as a generalization of a K-system in the case of positive entropy. We inve…
Assuming positive entropy we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for these actions.
We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map $f$ on a topological graph $G$ has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak's Möbius Disjointness Conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no $3$-scrambled tuple if and only if the p…
The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associ…
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.
We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.
We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a B…
We introduce a property of mutation loops, called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropical $\mathcal{X}$-transformation. A sign-stable mutation loop has a numerical invariant which we call the cluster stretch factor, in analogy with that of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor.
We introduce the algebraic entropy for continuous endomorphisms of locally linearly compact vector spaces over a discrete field, as the natural extension of the algebraic entropy for endomorphisms of discrete vector spaces. We show that the main properties of entropy continue to hold in the general context of locally linearly compact vector spaces, in particular we extend the Addition Theorem.
An exercise in experimental mathematics: calculation of the algebraic entropy of a map
本文通过一个射影空间自映射的代数熵计算实例，展示了导出递推方法在实验数学中的应用，并说明在缺乏完整证明时多种方法高度一致可支持对精确结果的信任。
We extend the definition of algebraic entropy to semi-discrete (difference-differential) equations. Calculating the entropy for a number of integrable and non integrable systems, we show that its vanishing is a characteristic feature of integrability for this type of equations.
This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim in [4] as one of its subfamilies.
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.
In arXiv:1109.6438v1 [math.AG] we introduced and studied a notion of algebraic entropy. In this paper we will give an application of algebraic entropy in proving Kunz Regularity Criterion for all contracting self-maps of finite length of Noetherian local rings in arbitrary characteristic. Some conditions of Kunz Criterion have already been extended to the general case by Avramov, Iyengar and Miller in arXiv:math/0312412v2 [math.AC], using different methods.
The notion of intrinsic algebraic entropy of an endomorphism of a given Abelian group has been recently introduced in [D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Intrinsic algebraic entropy, J. Pure Appl. Algebra 219 (2015) 2933-2961]. In this short note we provide a correct argument to prove one of the basic properties of the intrinsic algebraic entropy: the Logarithmic Law. In fact, this property was correctly stated in [op. cit.] but, as we will show with an explicit counterexample, the original proof contains a flaw.
We study the algebraic entropy of continuous endomorphisms of compactly covered, locally compact, topologically quasihamiltonian groups. We provide a Limit-free formula which helps us to simplify the computations of this entropy. Moreover, several Addition Theorems are given. In particular, we prove that the Addition Theorem holds for endomorphisms of quasihamiltonian torsion FC-groups (e.g., Hamiltonian groups).
We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on X, it is shown that the entropy of the a…
Algebraic entropy of birational automorphism groups in families of hyper-Kähler manifolds
本文研究射影双Kähler流形族中双有理自同构群的代数熵行为，探讨其性质与变化。
The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.
The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of locally compact groups, paying special attention to the case of compact and discrete groups respectively. The basic properties of these entropies, as well as many examples, are recalled. Also new entropy functions are proposed, as well as generalizations of several known definitions and results. Furthermore we give some connections with other topics in Math…
Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group
本研究提出统一框架关联代数作用与拓扑熵相关结论，引入马尔可夫性质证明熵与渐近对的关联，并借助独立熵对刻画两类顺从群有限表现膨胀代数作用的同宿群。
In 1965 Adler, Konheim and McAndrew defined the topological entropy for continuous self-maps of compact spaces. Topological entropy is very well-understood for endomorphisms of compact Abelian groups. A fundamental result in this context is the so-called Yuzvinski Formula, showing that the value of the topological entropy of a full solenoidal automorphism coincides with the Mahler measure of its characteristic polynomial. In two papers of 1979 and 1981 Peters gave a different definition of entropy for automorphisms of locally compact Abelian groups. This notion has been appropriately modified …
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…
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.
Bohr chaoticity of principal algebraic actions and Riesz product measures
该研究提出拓扑共轭不变的Bohr混沌性概念（强于正熵），证明正熵主代数$\u2124$作用均为Bohr混沌，且$d\ge2$时正熵主代数$\u2124^d$作用在可和同宿点存在条件下也满足该性质。
In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first $\ell^2$-Betti number of the group. More generally, we study coboun…
We introduce and study a notion of algebraic entropy for self-maps of finite length of Noetherian local rings, and develop its properties. We show that it shares the standard properties of topological entropy. For finite self-maps we explore the connection between the degree of the map and its algebraic entropy, when the ring is a Cohen-Macaulay domain. As an application of algebraic entropy, we give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity.
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…
Let G be a sofic group and X a compact group that G acts on by automorphisms. Using (and reformulating) the notion of doubly-quenched convergence developed by Austin, we show that in many cases the topological and the measure-theoretic entropy with respect to the Haar measure of this action agree. Our method of proof recovers all known examples. Moreover, the proofs are direct and do not go through explicitly computing the measure-theoretic or topological entropy.
The algebraic entropy of one-dimensional finitary linear cellular automata
本文从代数角度研究一维有限线性细胞自动机，证明其对偶为经典线性自动机，给出可逆条件，并计算代数熵，引入次数概念以描述熵。
Computing Garsia Entropy for Bernoulli Convolutions with Algebraic Parameters
该研究引入含特定代数整数的参数空间与递增分段连续函数序列给出伯努利卷积Garsia熵下界，证明部分开区域内对应伯努利卷积的Hausdorff维数为1。
We introduce the notion of intrinsic semilattice entropy $\widetilde h$ in the category $\mathcal L_{qm}$ of generalized quasimetric semilattices and contractive homomorphisms. By using appropriate categories $\mathfrak X$ and functors $F:\mathfrak X\to\mathcal L_{qm}$ we find specific known entropies $\widetilde h_\mathfrak X$ on $\mathfrak X$ as intrinsic functorial entropies, that is, as $\widetilde h_\mathfrak X=\widetilde h\circ F$. These entropies are the intrinsic algebraic entropy, the algebraic and the topological entropies for locally linearly compact vector spaces, the topological e…
An algebraic $Z^{d}$-action is an action of $Z^{d}$ on a compact abelian group $X$ by automorphisms of $X$. We prove that for $d \ge 8$, there exist mixing zero entropy algebraic $Z^{d}$-actions which do not exhibit isomorphism rigidity property.
We prove a max-min theorem for weak containment in the context of algebraic actions. Namely, we show that given an algebraic action of $G$ on $X,$ there is a maximal, closed $G$-invariant subgroup $Y$ of $X$ so that the action of $G$ on $Y$ is weakly contained in a Bernoulli shift. This subgroup is also the minimal subgroup so that any action weakly contained in a Bernoulli shift is $G\curvearrowright X/Y$-ergodic "in the presence of $G\curvearrowright X$". We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal …
Various limit-free formulas are given for the computation of the algebraic and the topological entropy, respectively in the settings of endomorphisms of locally finite discrete groups and of continuous endomorphisms of totally disconnected compact groups. As applications we give new proofs of the connection between the algebraic and the topological entropy in the abelian case and of the connection of the topological entropy with the finite depth for topological automorphisms.
For a positive integer $M$ and a real base $q\in(1,M+1]$, let $\mathcal{U}_q$ denote the set of numbers having a unique expansion in base $q$ over the alphabet $\{0,1,\dots,M\}$, and let $\mathbf{U}_q$ denote the corresponding set of sequences in $\{0,1,\dots,M\}^{\mathbb{N}}$. Komornik et al. [Adv. Math. 305 (2017), 165--196] showed recently that the Hausdorff dimension of $\mathcal{U}_q$ is given by $h(\mathbf{U}_q)/\log q$, where $h(\mathbf{U}_q)$ denotes the topological entropy of $\mathbf{U}_q$. They furthermore showed that the function $H: q\mapsto h(\mathbf{U}_q)$ is continuous, nondecr…
We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, …
Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive act…
We investigate rigidity of measurable structure for higher rank abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a 0 entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.
We show that algebraic dynamical systems with entropy rank one have uniformly exponentially many periodic points in all directions.
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。
Let X_1 and X_2 be mixing connected algebraic dynamical systems with the Descending Chain Condition. We show that every equivariant continuous map X_1 to X_2 is affine (that is, X_2 is topologically rigid) if and only if the system X_2 has finite topological entropy.
In this article we discuss relations between algebraic and dynamical properties of non-cyclic semigroups of rational maps.
The aim of this work is to study a kind of refinement of the entropy conjecture, in the context of partially hyperbolic diffeomorphisms with one dimensional central direction, of d-dimensional torus. We start by establishing a connection between the unstable index of hyperbolic sets and the index at algebraic level. Two examples are given which might shed light on which are the good questions in the higher dimensional center case.
We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.
We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak$^{*}$-limit of mea…
We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show isomorphism rigidity for such actions, and to provide examples of non-isomorphic Z^3-actions with all their Z^2-sub-actions isomorphic. The proofs use lexicographic half-space entropies and total ergodicity along critical directions.
We survey some of the known criteria for expansiveness of principal algebraic actions of countably infinite discrete groups. In the special case of the discrete Heisenberg group we propose a new approach to this problem based on Allan's local principle. Furthermore, we present a first example of an absolutely summable homoclinic point for a nonexpansive action of the discrete Heisenberg group and use it to construct an equal-entropy symbolic cover of the system.
On the support of measures of large entropy for Hénon-Sibony maps
研究Hénon-Sibony映射的大熵不变测度的支撑集，证明若测度熵大于某阈值则支撑在Julia集上。
In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system $(f,X,Ω)$. The main idea is to make use - in addition to invariant compact subspaces of $(X,Ω)$ - of compactifications of étale covers $π:(f',X',Ω')\rightarrow (f,X,Ω)$, that is $π\circ f'=f\circ π$ and the fibers of $π$ are all finite. We prove some basic results and propose a conjecture, who…
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [4]. Later, this result was extended to all abelian groups [3] and, recently, to all torsion finitely quasihamiltonian groups [7]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [8]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.
We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invert…
Algebraic expansivity on abelian groups
本文研究了阿贝尔群上端态射的代数扩张性，探讨了其基本性质、与Weiss代数熵的关系，并证明正向扩张满射仅限于有限系统，同时通过庞特里亚金对偶性展示了其与拓扑动力学的联系。
In a previous paper the authors developed an operator-algebraic approach to Lewis Bowen's sofic measure entropy that yields invariants for actions of countable sofic groups by homeomorphisms on a compact metrizable space and by measure-preserving transformations on a standard probability space. We show here that these measure and topological entropy invariants both coincide with their classical counterparts when the acting group is amenable.
Motivated by the coincidence of topological entropies the connection between abelian sandpiles and harmonic models was established by K. Schmidt and E. Verbitskiy (2009). The dissipative sandpile models were shown to be symbolic representations of algebraic $Z^d$-actions of the harmonic models. Both models are determined by so-called simple sandpile polynomials. We extend this result to arbitrary sandpile polynomials. Moreover, we show that any sandpile model determined by a factor of a sandpile polynomial acts as an equal entropy cover of the corresponding algebraic model. For a special class…
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