Richard Miles, Matthew Staines, Thomas Ward
We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given and the problem of determining the range of certain invariants of topological conjugacy is discussed. Several new results and old and new open problems are described.
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We prove that the restriction of a probability measure invariant under a nonhyperbolic, ergodic and totally irreducible automorphism of a compact connected abelian group to the leaves of the central foliation is severely restricted. We also prove a topological analogue of this result: the intersection of every proper closed invariant subset with each central leaf is compact.
We suggest a combinatorial classification of metric filtrations in measure spaces; a complete invariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies of the group~$\mathbb Z$. In turn, the notion of combinatorial scheme is a source of new metric invariants of automorphisms approximated via basic filtrations. We construct a universal graph endowed with an adic structure such that every automorphism can be realized in its path space.
The role of generalized Bowen-Franks groups (BF-groups) as topological conjugacy invariants for $\mathbb{T}^{n}$ automorphisms is studied. Using algebraic number theory, a link is established between equality of BF-groups for different automorphisms ($BF$-equivalence) and an identical position in a finite lattice ($\mathcal{L}$-equivalence). Important cases of equivalence of the two conditions are proved. Finally, a topological interpretation of the classical BF-group $\mathbb{Z}% ^{n}/\mathbb{Z}^{n}(I-A)$ in this context is presented.
Mixing and equipartition for automorphism invariant processes on regular trees
针对正则树上保奇偶性树自同构不变的有限状态过程，证明其遍历过程在水平球面内沿球面和球几乎处处等分，并在定量混合条件下得到偶数半径度量球面的香农-麦克米兰-布莱曼定理。
We show that a collection of generalized Bowen-Franks group, what we call the principal Bowen-Franks $R$-modules, form a complete set of $R$-module invariants for the equivalence relation of profinite conjugacy for similar hyperbolic toral automorphisms. We also show that these principal generalized Bowen-Franks $R$-modules are the principal invariants in a large class of generalized Bowen-Franks $R$-module invariants for similar hyperbolic automorphisms. We further show how the conjugacy invariant of cyclicity, when applied to hyperbolic toral automorphisms, is realized for profinite conjugac…
We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator modulo the sigma-ideal generated by wandering sets. This implies that if the entropies of invariant probability measures of a Borel system are all less than log(k), then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.
We construct an example of a birational transformation of a rational threefold for which the first and second dynamical degrees coincide and are $>1$, but which does not preserve any holomorphic (singular) foliation. In particular, this provides a negative answer to a question of Guedj. On our way, we develop several techniques to study foliations which are invariant under birational transformations.
If $α$ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral automorphism), then $α$ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in $X$. In spite of this we are able to construct a symbolic space $V$ and a class of shift-invariant probability measures on $V$ each of which corresponds to an $α$-invariant probability measure on $X$. Moreover, every $α$-invariant probability measure on $X$ arises essentially in this way. The last part of the pa…
The Dynamical Degrees of Rational Surface Automorphisms
本文研究有理曲面自同构动力次数与 $E_n$ Coxeter 群元素谱半径的关系，通过为不可实现元素构造同谱半径的可实现元素，证明两集合完全相同。
For a $\mathbb{Z}^d$-action $α$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when $α$ is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind's conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point cou…
We construct automorphisms of $\mathbb{C}^2$, and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank 1. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):\mathbb{C}^2\rightarrow\mathbb{C}$ holomorphic.
Invariant complex structures for affine automorphisms: a cocycle viewpoint
本文证明紧复流形上的全纯微分同胚若双Lipschitz共轭于齐性空间上的遍历仿射自同构，则共轭光滑；若该自同构弱混合，则诱导复结构左不变。应用包括全纯Anosov微分同胚的正则性提升及高秩交换Anosov作用的刚性定理的全纯类比。
We examine invariant nonrecurrent Fatou components of automorphisms of $\mathbb{C}^2$ in the case where all limit maps are constant. We show that except in special cases there cannot be more than one such limit map. We also briefly examine such Fatou components where the limit maps may be nonconstant. Lastly we present a few examples of such Fatou components.
For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic le…
We describe all the dynamical degrees of automorphisms of hyperkähler manifolds in terms of the first dynamical degree. We also present two explicit examples of different geometric flavours.
We classify invariant probability measures for non-elementary groups of automorphisms, on any compact Kähler surface X, under the assumption that the group contains a so-called "parabolic automorphism". We also prove that except in certain rigid situations known as Kummer examples, there are only finitely many invariant, ergodic, probability measures with a Zariski dense support. If X is a K3 or Enriques surface, and the group does not preserve any algebraic subset, this leads to a complete description of orbit closures.
We study the possible dynamical degrees of automorphisms of the affine space $\mathbb{A}^n$. In dimension $n=3$, we determine all dynamical degrees arising from the composition of an affine automorphism with a triangular one. This generalises the easier case of shift-like automorphisms which can be studied in any dimension. We also prove that each weak Perron number is the dynamical degree of an affine-triangular automorphism of the affine space $\mathbb{A}^n$ for some $n$, and we give the best possible $n$ for quadratic integers, which is either $3$ or $4$.
We classify complex projective surfaces with an automorphism of positive entropy for which the unique invariant measure of maximal entropy is absolutely continuous with respect to Lebesgue measure.
We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an automorphism of positive entropy but no invariant curve. The Main Theorem: If $f_{a,b}$ is an automorphism with an invariant curve and positive entropy, then either (1) $(a,b)$ is real, and the restriction of $f$ to the real points has maximal entropy, or (2) $f_{a,b}$ has a rotation (Siegel) domain.
The stabilized automorphism group of odometers and of Toeplitz subshifts
本文刻画了里程表与Toeplitz子移位的稳定自同构群，证明其不变性质，并指出对无挠里程表该群是完全的构同不变元。
We show that if $r\geq 3$ and $α$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $α$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.
We prove that for a polynomial diffeomorphism of C^2 , the support of any invariant measure, apart from a few obvious cases, is contained in the closure of the set of saddle periodic points.
The polynomial automorphisms of the affine plane have been studied a lot: if $f$ is such an automorphism, then either $f$ preserves a rational fibration, has an uncountable centralizer and its first dynamical degree equals $1$, or $f$ preserves no rational curves, has a countable centralizer and its first dynamical degree is $>1$. In higher dimensions there is no such description. In this article we study a family $(Ψ_α)_α$ of polynomial automorphisms of $\mathbb{C}^3$. We show that the first dynamical degree of $Ψ_α$ is $>1$, that $Ψ_α$ preserves a unique rational fibration and has an uncount…
Quadratic automorphisms of $\mathbb C^3$ are classified up to affine conjugacy into seven classes by Forn$æ$ss and Wu. Five of them contain irregular maps with interesting dynamics. In this paper, we focus on the maps in the fifth class and make some dynamical estimates for these maps.
Lind and Schmidt have shown that the homoclinic group of a cyclic $\Z^k$ algebraic dynamical system is isomorphic to the dual of the phase group. We show that this duality result is part of an exact sequence if $k=1$. The exact sequence is a well known algebraic object, which has been applied by Schmidt in his work on rigidity. We show that it can be derived from dynamical considerations only. The constructions naturally lead to an almost 1-1-coding of certain Pisot automorphisms by their associated $β$-shift, generalizing similar results for Pisot automorphisms of the torus.
We give a way to construct group of pseudo-automorphisms of rational varieties of any dimension that fix pointwise the image of a cubic hypersurface of $P^n. These group are free products of involutions, and most of their elements have dynamical degree >1. Moreover, the Picard group of the varieties obtained is not big, if the dimension is at least 3. We also answer a question of E. Bedford on the existence of birational maps of the plane that cannot be lifted to automorphisms of dynamical degree >1, even if we compose them with an automorphism of the plane.
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques to show that automorphism groups of minimal zero entropy shifts are sofic.
For any given Salem number, we construct an automorphism on a simple abelian variety whose first dynamical degree is the square of the Salem number. Our construction works for both simple abelian varieties with totally indefinite quaternion multiplication and for simple abelian varieties of the second kind. We then give a complete classification of the dynamical degree sequences for abelian varieties of dimension at most four and obtain an ergodic result for sequences of pullbacks of forms.
Non-archimedean and hybrid dynamics of regular polynomial automorphisms
本文利用混合空间技术，证明正则多项式自同构解析族在退化时的不变测度弱极限等于非阿基米德域上诱导自同构的不变测度，推广了Hénon映射情形的先前结果。
We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and regularity of solutions of the cohomological equation for such automorphisms. Our method is based on the quantitative equidistribution results for polynomial maps combined with Diophantine estimates.
Dynamical degrees of affine-triangular automorphisms in dimension four
本文证明了四维仿射-三角自同构的动力次数是次数不超过4的代数整数，并在char(k)≠2时推广至四维二次自同构，部分肯定地回答了DF21中的猜想。
Dynamical Mordell-Lang problem for automorphisms of surfaces in positive characteristic
本文在正特征下研究了曲面自同构的动力学Mordell-Lang问题，通过分析动力系统与代数几何的相互作用，揭示了有限映射下的轨道结构与代数簇性质之间的联系。
In connection with Rokhlin's question on an automorphism with a homogeneous nonsimple spectrum, we indicate a class of measure-preserving maps $T$ such that $T\times T$ has a homogeneous spectrum of multiplicity 2. The automorphisms in question satisfy the condition $σ\astσ\perp σ$, where $σ$ is the spectral measure of $T$. We also show that there is a mixing automorphism possessing the above properties and their higher order analogs. This work has been published in the pilot issue of Selected Russian Mathematics, Vol.1 (1999), no.1, 13-24.
Dynamical Tits alternative for groups of almost automorphisms of trees
本文证明了局部有限树的几乎自同构群满足动力学 Tits 替换，将其在边界上的作用分为包含非交换自由群或保持概率测度两类，推广了关于 Thompson 群 $V$ 的相关结果。
We show that in the neighborhood of each ``finite type'' singular orbit of a real analytic integrable dynamical system (hamiltonian or not) there is a real analytic torus action which preserves the system and which is transitive on this orbit. We also show that the local automorphism group of the system near such an orbit is
We prove the existence of automorphisms of $\mathbb C^k$, $k\ge 2$, having an invariant, non-recurrent Fatou component biholomorphic to $\mathbb C \times (\mathbb C^\ast)^{k-1}$ which is attracting, in the sense that all the orbits converge to a fixed point on the boundary of the component. Such a Fatou component also avoids $k$ analytic discs intersecting transversally at the fixed point. As a corollary, we obtain a Runge copy of $\mathbb C \times (\mathbb C^\ast)^{k-1}$ in $\mathbb C^k$.
Density of finitely supported invariant measures for automorphisms of compact abelian groups
本文证明满足降链条件的紧可度量阿贝尔群自同构的有限支撑不变测度在弱-*拓扑下稠密，并利用specification性质变体得到Hilbert-Schmidt稳定性与Livshitz型定理等推论。
We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $\mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a si…
Let $X$ be a compact Kähler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $λ_1(f)^2>λ_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be…
Let $X$ be a projective variety and $σ$ a wild automorphism on $X$, i.e., whenever $σ(Z) = Z$ for a non-empty Zariski-closed subset $Z$ of $X$, we have $Z = X$. Then $X$ is conjectured to be an abelian variety with $σ$ of zero entropy (and proved to be so when ${\rm dim} \, X \le 2$) by Z. Reichstein, D. Rogalski and J. J. Zhang in their study of projectively simple rings. This conjecture has been generally open for more than a decade. In this note, we confirm this original conjecture when ${\rm dim} \, X \le 3$ and $X$ is not a Calabi-Yau threefold, and also show that $σ$ is of zero entropy w…
Uniform boundedness on rational maps with automorphisms
针对自同构群含 d 阶循环群、次数 d≥2 的有理映射族，论文证明其某一子族满足动力学一致有界性原理。
We examine the completely isometric automorphisms of a free product of noncommutative disc algebras. It will be established that such an automorphism is given simply by a completely isometric automorphism of each component of the free product and a permutation of the components. This mirrors a similar fact in topology concerning biholomorphic automorphisms of product spaces with nice boundaries due to Rudin, Ligocka and Tsyganov. This paper is also a study of multivariable dynamical systems by their semicrossed product algebras. A new form of dynamical system conjugacy is introduced and is sho…
We say (W, \{ϕ_1,..., ϕ_t\}) is a polarizable dynamical system of several morphisms if ϕ_i are endomorphisms on a projective variety $W$ such that \bigotimes ϕ_i^*L is linearly equivalent to L^q} for some ample line bundle L on W and for some q>t. If q is a rational number, then we have the equidistribution of small points of given dynamical system because of Yuan's work. As its application, we can build a polarizable dynamical system of an automorphism and its inverse on $K3$ surface and show its periodic points are equidistributed.
This sequel to our previous paper [MS11b] continues the study of topological contact dynamics and applications to contact dynamics and topological dynamics. We provide further evidence that the topological automorphism groups of a contact structure and a contact form are the appropriate transformation groups of contact dynamical systems. The article includes an examination of the groups of time-one maps of topological contact and strictly contact isotopies, and the construction of a bi-invariant metric on the latter. Moreover, every topological contact or strictly contact dynamical system is a…
We exhibit continua on two different growth scales in the dynamical Mertens' theorem for ergodic automorphisms of one-dimensional solenoids.
We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of C^2. Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.
Let $S=\{p_1, \dots, p_r,\infty\}$ for prime integers $p_1, \dots, p_r.$ Let $X$ be an $S$-adic compact nilmanifold, equipped with the unique translation invariant probability measure $μ.$ We characterize the countable groups $Γ$ of automorphisms of $X$ for which the Koopman representation $κ$ on $L^2(X,μ)$ has a spectral gap. More specifically, we show that $κ$ does not have a spectral gap if and only if there exists a non-trivial $Γ$-invariant quotient solenoid (that is, a finite-dimensional, connected, compact abelian group) on which $Γ$ acts as a virtually abelian group.
We generalise a recent example by F. Bracci, J. Raissy and B. Stensønes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every sta…
We describe an explicit method for constructing pseudo-automorphisms of a space $X$ which is obtained by blowing up points of $P^k$ (or a product $P^k \times \cdots \times P^k$). The centers of blowup are chosen to lie on an elliptic normal curve and are determined using the arithmetic on the curve. These pseudo-automorphism have dynamical degree greater than $1$.
Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{ϕ\in\text{PGL}_3(K): ϕ^{-1}\circ f\circϕ=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.
The features of the paper are these: 1) we discuss {\bf especially} quadratic (alias bilinear) Bose-Hamiltonians, the related Bogoliubov transformations and {\bf especially} quasi-free-like (alias coherent or Fock-like) states 2) we discuss {\bf any} quadratic Bose-Hamiltonians and Bogoliubov transformations, whether diagonalizable or not, whether proper or improper, and {\bf arbitrary} quasi-free-like states, whether regular or non-regular they are 3) we associate notions and terms of the CCRs theory {CCRs = Canonical Commutation Relations} with notions and terms of the indefinite inner produ…
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.
Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a time-reversal symmetry; in particular the Jacobian $\mathrm{Jac}(f)$ must be a root of unity. As a step towards this conjecture, we prove that the Jacobian of $f$ and all its Galois conjugates lie on the unit circle in the complex plane. Under mild additional assumptions we are able to conclude that indeed $\mathrm{Jac}(f)$ is a root of unity. We use these resu…
We consider pairs of toral automorphisms (A,B) satisfying an invariant cone property. At each iteration, A acts with probability p and B with probability 1-p. We prove exponential decay of correlations for a class of Holder continuous observables.
We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $…
We show that if $ϕ: X \to X$ is an automorphism of a smooth projective variety and $D \subset X$ is an irreducible divisor for which the set of $d$ in $D$ with $ϕ^n(d)$ in $D$ for some nonzero $n$ is not Zariski dense, then $(X, ϕ)$ admits an equivariant rational fibration to a curve. As a consequence, we show that certain blowups (e.g. blowups in high codimension) do not alter the finiteness of $\textrm{Aut}(X)$, extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety …
We equate dynamical properties (e.g., positive entropy, existence of a periodic curve) of complex projective surface automorphisms with properties of the pull-back actions of such automorphisms on line bundles. We use the properties of the cohomological actions to describe the measures of maximal entropy for automorphisms with positive entropy.
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