Conjugacy Invariants Research Articles

A duality for endomorphisms of von Neumann algebras

We consider endomorphisms of von Neumann algebras: Let M be a von Neumann algebra, represented on a Hilbert space ℋ, and let M′ be the corresponding commutant. Let α∈End(M) be given, and suppose M has a cyclic vector in ℋ, such that the corresponding state leaves α invariant. Then there is a ‘‘dual’’ completely positive mapping β on M′ which we find and describe: Each of the two α and β has an associated spectral group, and we show that the group for β is contained in that for α. We consider the following three restrictions on α: i) α is a shift on M, ii) α is strongly ergodic, and iii) α is ergodic. We give spectral theoretic conditions on α (using the two groups described above) to fall into each of the three classes. We also show that the two groups are conjugacy invariants, and we discuss the case of cocycle conjugacy.

Topologie d'un polynôme de deux variables complexes au voisinage de l'infini

We give a complete system of invariants for the topological conjugacy of polynomials of ℂ 2 outside a big enough compact set in the two possible versions: as foliations (forgetting the values of the fibers) and as functions. These invariants are described as a weighted and colored tree, that is obtained after reduction of singularities of the polynomial in the line of infinity. We give a regularity criteria for the values of a polynomial and a description of the topology of its fibers used in the construction of the topological conjugacy from the tree.

On some invariants of conjugacy of disjoint iteration groups

We give a necessary and sufficient condition for conjugacy of disjoint iteration groups of continuous functions defined on open real intervals. Moreover, we determine all homeomorphisms γ such that γ o ƒt = g t o γ, t ∈ R, where {ƒt, t R} and {g1, t ∈ R} are given disjoint iteration groups.

Pseudo-periodic homeomorphisms and degeneration of Riemann surfaces

We will announce two theorems. The first theorem will classify all topological types of degenerate fibers appearing in one-parameter families of Riemann surfaces, in terms of "pseudoperiodic" surface homeomorphisms. The second theorem will give a complete set of conjugacy invariants for the mapping classes of such homeomorphisms. This latter result implies that Nielsen’s set of invariants [Surface transformation classes of algebraically finite type, Collected Papers 2, Birkhäuser (1986)] is not complete.

An Inclusion of Type III Factors with Index 4 Arising from an Automorphism

We consider an inclusion of type \mathrm{III} factors with index 4 arising from an automorphism. It has the extended Coxeter–Dynkin diagram \tilde A or A_{∞,∞} as the principal graph. We show the equivalence between classification of automorphisms and that of the corresponding subfactors, and compute conjugacy invariants for the subfactors. As an application, we show the existence of a pair of type \mathrm{III}_1 factors which does not split into a type \mathrm{II}_1 inclusion.

Inclusions of Type III Factors Arising from Finite Group Actions

We consider inclusions of type III factors arising from finite group actions. We show the relation between actions and the corresponding subfactors, and compute conjugacy invariants for the subfactors.

Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems

We give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous, then it is as smooth as the maps. We furthermore prove generalizations of these facts for non-uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions.

Conjugacy invariants of möbius transformations

We define conjugacy invariants of Clifford matrices, which have been introduced by Ahlfors to generalize the Mobius transformation groups PSL (2,R) and PSL (2, C) to higher dimensions. Our conjugacy invariants are a generalization of the trace square functions. We also show explicitly how these invariants determine the conjugacy class of a Mobius transformation.

Homoclinic points and moduli

AbstractIn this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.

Invariants for smooth conjugacy of hyperbolic dynamical systems. IV

We show that if twoC ∞ transitive Anosov flows in a three-dimensional manifold are topologically conjugate and the Lyapunov exponents on corresponding periodic orbits agree, then the conjugating homeomorphism isC ∞.

Invariants for smooth conjugacy of hyperbolic dynamical systems. I

Invariants for smooth conjugacy of hyperbolic dynamical systems. I

On moduli of stability of two-dimensional vector fields

In this paper the author proves that, on a compact connected and orientable two-dimensionalC∞ manifoldM, the gradient of aC3-Morse Function has finite modulus of stability under conjugacy and modulus zero under topological equivalence. It is also proved that generically the modulus of stability under conjugacy of a graph of aC2 vector field on the plane is at least twice the number of its saddles. Some new conjugacy invariants arise in the proofs of these results.

Isometries of ergodic Hardy spaces

The isometries between ergodic Hardy spaces are identified. As a consequence it is shown that the isometry class of an ergodic Hardy space associated with an ergodic, measure preserving flow constitutes an essentially complete set of conjugacy invariants for the flow.