Some remarks on homoclinic groups of hyperbolic toral automorphisms

The homoclinic group (an invariant with respect to topological conjugacy) for hyperbolic toral automorphisms is determined. Certain conditions are given for conjugacy of a homeomorphism of a compact space to hyperbolic toral automorphism. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1994, pp. 140–147. This paper is partially supported by Russian Foundation for Basic Research, grant 94-01-00921. Translated by V. V. Sadovskaya.     

CC-groups with periodic central factor

A group G is said to be a group with Černikov conjugacy classes or a CC-group if it induces on the normal closure of each one of its elements a group of automorphisms which is a Černikov group, that is, a finite extension of an abelian group satisfying the minimal condition on subgroups. This concept is a natural extension of that an FC-group, that is, a group in which every element has a finite number of conjugates. It is known that if G is an FC-group then the central factor G/Z(G) is periodic. This result does not hold for CC-groups and in this paper we study CC-groups G in which the central factor G/Z(G) is periodic, a finiteness condition which has a deep influence on the structure of the group G. In particular, we characterize those CC-groups as above that are FC-groups by imposing some additional conditions on their structure. This research has been supported by DGICYT (Spain) PS88-0085     

A lattice point problem on the regular tree

Huber (1956) [8] considered the following problem on the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x∈H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree.     

On conjugacy in groups of finite-state automorphisms of rooted trees

We show that the conjugacy of elements of finite order in the group of finite-state automorphisms of a rooted tree is equivalent to their conjugacy in the group of all automorphisms of the rooted tree. We establish a criterion for conjugacy between a finite-state automorphism and the adding machine in the group of finite-state automorphisms of a rooted tree of valency 2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1357–1366, October, 2008.     

Markov shift in non-commutative probability

We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of ‘quantum detailed balance’ introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.     

Rigidity of measurable structure for \mathbb Z^d-actions by automorphisms of a torus

We show that for certain classes of actions of , by automorphisms of the torus any measurable conjugacy has to be affine, hence measurable conjugacy implies algebraic conjugacy; similarly any measurable factor is algebraic, and algebraic and affine centralizers provide invariants of measurable conjugacy. Using the algebraic machinery of dual modules and information about class numbers of algebraic number fields we construct various examples of -actions by Bernoulli automorphisms whose measurable orbit structure is rigid, including actions which are weakly isomorphic but not isomorphic. We show that the structure of the centralizer for these actions may or may not serve as a distinguishing measure-theoretic invariant. Received: March 12, 2002     

Polarities and unitals in the Coulter–Matthews planes

For a class of finite shift planes introduced by Coulter and Matthews, we give a set of representatives for the isomorphism types, determine all automorphisms and describe all polarities explicitly. The planes in question are the only known examples of finite shift planes that are not translation planes. Each non-desarguesian Coulter–Matthews plane admits precisely two conjugacy classes of orthogonal polarities. In addition, each Coulter–Matthews plane of square order admits exactly one conjugacy class of unitary polarities. We prove that most of the corresponding unitals are not classical.     

Sur la classification des automorphismes périodiques des C-algèbres UHF

We obtain a complete classification up to conjugacy and up to outer conjugacy of finite tensor product type automorphisms of UHF C^*-algebras which are periodic of period N (N prime).     

Sur la classification des automorphismes périodiques des C1-algèbres UHF

We obtain a complete classification up to conjugacy and up to outer conjugacy of finite tensor product type automorphisms of UHF C¹-algebras which are periodic of period N (N prime).     

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)^Φ its field of invariants, i.e., the set of rational functions f on X such that f o Φ = f. Let n(Φ) be the transcendence degree of k(X)^Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕ_g, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.     

Conjugacy classes of non-connected semisimple algebraic groups

Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G is isomorphic to (T/(1 − τ )(T))/W^τ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of the Ph.D thesis of the author, cf. [15].     

On graphs related to conjugacy classes of groups

LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.     

N/V-limit for stochastic dynamics in continuous particle systems

We provide an N/V-limit for the infinite particle, infinite volume stochastic dynamics associated with Gibbs states in continuous particle systems on ℝ ^d ,d≥1. Starting point is an N-particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset Λ⊂ℝ ^d with finite volume (Lebesgue measure) V=|Λ|<∞. The aim is to approximate the infinite particle, infinite volume stochastic dynamic by the above N-particle dynamic in Λ as N→∞ and V→∞ such that N/V→ρ, where ρ is the particle density. First we derive an improved Ruelle bound for the canonical correlation functions under an appropriate relation between N and V. Then tightness is shown by using the Lyons–Zheng decomposition. The equilibrium measures of the accumulation points are identified as infinite volume canonical Gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle, infinite volume stochastic dynamics via the associated martingale problem. Assuming a property closely related to Markov uniqueness and weaker than essential self-adjointness, via Mosco convergence techniques we can identify the accumulation points as Markov processes and show uniqueness. I.e., all accumulation corresponding to one invariant canonical Gibbs measure coincide. The proofs work for general repulsive interaction potentials ϕ of Ruelle type and all temperatures, densities, and dimensions d≥1, respectively. ϕ may have a nontrivial negative part and infinite range as e.g. the Lennard–Jones potential. Additionally, our result provides as a by-product an approximation of grand canonical Gibbs measures by finite volume canonical Gibbs measures with empty boundary condition.     

Classification of disordered tilings

A tilingT is a disordered realization of a periodic tilingP with symmetry group Γ if we can map the complement of a compact set ofT onto the quotientP/Γ in such a way that this map respects the features of the tilingT andP. We show that the global type of a 2-dimensional tilingT is determined by the periodic tilingP it is a disordered realization of, a conjugacy class of Γ which can be associated toT and a winding number. In some cases, we need in addition some kind of orientation. For higher-dimensional tilings of spaces which are simply connected at infinity, e.g. ℝ ⁿ withn≥3, the associated periodic tiling alone is sufficient.     

Shift Automorphisms of the Hyperfinite Factor

For a large class of shift transformations of a LEBESGUE measure space (they have to fulfil some mixing condition) we construct automorphisms of the hyperfinite factor of type II₁. The CONNES -STØRMER entropy of the resulting automorphisms equals the measure theoretic entropy of the corresponding shift transformations. Two such automorphisms are conjugate if the conjugacy between the original measure space shifts can be given by a code with finite expected code length.     

Tight upper bounds on the number of invariant components on translation surfaces

An abelian differential on a surface defines a flat metric and a vector field on the complement of a finite set of points. The vertical flow that can be defined on the surface has two kinds of invariant closed sets (i.e. invariant components) — periodic components and minimal components. We give upper bounds on the number of minimal components, on the number of periodic components and on the total number of invariant components in every stratum of abelian differentials. We also show that these bounds are tight in every stratum.     

Hermitian symmetric spaces and Kahler rigidity

We characterize irreducible Hermitian symmetric spaces which are not of tube type, both in terms of the topology of the space of triples of pairwise transverse points in the Shilov boundary, and of two invariants which we introduce, the Hermitian triple product and its complexification. We apply these results and the techniques introduced in [6] to characterize conjugacy classes of Zariski dense representations of a locally compact group into the connected component G of the isometry group of an irreducible Hermitian symmetric space which is not of tube type, in terms of the pullback of the bounded Kahler class via the representation. We conclude also that if the second bounded cohomology of a finitely generated group Γ is finite dimensional, then there are only finitely many conjugacy classes of representations of Γ into G with Zariski dense image. This generalizes results of [6].     

Generalized Kac problem for Bose gas in polygonal region

The paper considers interacting Bose gas in a polygonal domain and studies the asymptotics of the log-partition function in its Feynman-Kac representation as the domain is delated to infinity. It is proved that for repulsive interaction with power decay at infinity the asymptotics of the log-partition function determines the area of the domain, the length of its boundary and the constant term defined by the angles of the polygon. This is a natural generalization of the Kac’s famous problem on computing the asymptotics of the partition function Tre ^βδ, where δ is the Dirichlet Laplacian for a polygonal domain.     

The information cocycle andɛ-bounded codes

We shall prove here that Bowen’s bounded codes lead to a cocycle-coboundary equation which can be exploited in various ways: through central limit theorems, through the related information variance or through a certain group invariant. Another result which emerges is that it is impossible to boundedly code two Markov automorphisms when one is of maximal type and the other is not. The functions which appear in the above cited cocycle-coboundary equation may belong to variousL ^p spaces. We devote a section to this problem. Finally we show that the information cocycle associated with small smooth partitions of aC ² Anosov diffeomorphism preserving a smooth probability is, in a sense, canonical.     

Trialitarian automorphisms of lie algebras

Over an algebraically closed field of characteristic zero simple Lie algebras admit outer automorphisms of order 3 if and only if they are of type D₄. Moreover, thereare two conjugacy classes of such automorphisms. Among orthogonal Lie algebras over arbitrary fields of characteristic zero, only orthogonal Lie algebras relative to quadratic norm forms of Cayley algebras admit outer automorphisms of order 3. We give a complete list of conjugacy classes of outer automorphisms of order 3 for orthogonal Lie algebras over arbitrary fields of characteristic zero. For the norm form of a given Cayley algebra, one class is associated with the Cayley algebra and the others with central simple algebras of degree 3 with involution of the second kind such that the cohomological invariant of the involution is the norm form.