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.     

On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by Aut _N ^M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut _N ^M (G), Aut _N ^M (G) ≤ Inn(G) or Inn(G) ≤ Aut _N ^M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.     

Differentiable rigidity for hyperbolic toral actions

Our aim is to extend existing results about differentiable rigidity of higher rank abelian actions by automorphisms of a torus. Previous proofs have required an assumption of semisimplicity, that is, that the action is by commuting diagonalizable matrices. Here we introduce a technique that utilizes the unipotent part of a non-semisimple action, which allows us to discard the semisimplicity assumption. In its place we will make a technical assumption that the spectrum of the action restricted to leaves of the coarse Lyapunov decomposition is sufficiently narrow.     

Common homoclinic points of commuting toral automorphisms

The points homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a ℤ²-action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same homoclinic points or have no homoclinic point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs. The second author would like to thank the Austrian Academy of Sciences and the Royal Society for partial support while this work was done.     

The prime orbit theorem for quasihyperbolic toral automorphisms

For a toral automorphism which is ergodic, but not necessarily hyperbolic, we derive asymptotic formulae for the number of closed orbits by analogy with the Prime Number Theorem. A new proof of the uniform distribution of periodic points is also given.     

Homoclinic processes and invariant measures for hyperbolic toral automorphisms

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L₂ with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.     

A note on periodic points for ergodic toral automorphisms

We prove that the periodic point measures are dense in the space or invariant measures for ergodic toral automorphisms.     

Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts

St.-Petersburg State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 22–27, July–September, 1992.     

Isomorphism rigidity of commuting automorphisms

Let , and let and be two zero-entropy -actions on compact abelian groups by commuting automorphisms. We show that if all lower rank subactions of and have completely positive entropy, then any measurable equivariant map from to is an affine map. In particular, two such actions are measurably conjugate if and only if they are algebraically conjugate.

     

The isomorphism problem for toral relatively hyperbolic groups

We provide a solution to the isomorphism problem for torsion-free relatively hyperbolic groups with abelian parabolics. As special cases we recover solutions to the isomorphism problem for: (i) torsion-free hyperbolic groups (Sela, [60] and unpublished); and (ii) finitely generated fully residually free groups (Bumagin, Kharlampovich and Miasnikov [14]). We also give a solution to the homeomorphism problem for finite volume hyperbolic n-manifolds, for n≥3. In the course of the proof of the main result, we prove that a particular JSJ decomposition of a freely indecomposable torsion-free relatively hyperbolic group with abelian parabolics is algorithmically constructible.     

Local automorphisms

We show that any linear map on a finite dimensional CSL algebra which at each point is equal to the value of some automorphism of is either an automorphism or can be factored as an automorphism and the transpose of a self-adjoint summand of . New examples of local mappings are constructed.