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.     

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.     

On commuting automorphisms of groups

This paper examines group automorphisms for which each element commutes with its image. These automorphisms do not necessarily form a subgroup of the automorphism group, but have a number of interesting properties, close to those of central automorphisms. Research of the first named author was supported by Grant SM 177 from Kuwait University Research Administration.     

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.

     

Local rigidity for certain groups of toral automorphisms

Let Γ = SL(n, ℤ) or any subgroup of finite index, n ≥ 4. We show that the standard action of Γ on ⁿ is locally rigid, i.e., every action of Γ on ⁿ by C^∞ diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a C^∞ diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, ℤ). Partially supported by NSF grant DMS9011749.     

Weight filtrations via commuting automorphisms

We consider a filtration of theK-theory space for a regular Noetherian ring proposed by Goodwillie and Lichtenbaum and show that its successive quotients are geometric realizations of explicit simplicial Abelian groups. The filtration in weightt involvest-tuples of commuting automorphisms of projectiveR-modules. It remains to show that the Adams operations act appropriately on the filtration.Supported by NSF grant DMS 90-02715. I thank Tom Goodwillie, Stephen Lichtenbaum, Friedhelm Waldhausen, Steven Landsburg, and Stephen Ullom for useful discussions and ideas.     

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.     

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.     

Common fixed points for R-weakly commuting in fuzzy metric spaces

In this paper, we study the concept of R-weakly commuting of type (A _g ) of Pathak et?al. (Bull Korean Math Soc 34:247?C257, 1997) in fuzzy metric spaces. We also establish the existence of common fixed point theorems by using the common limit in the range property and give an example to validate our the main results.     

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.     

Semicrossed products generated by two commuting automorphisms

In this paper, we study the semicrossed product of a finite dimensional C∗-algebra for two types of -actions, and identify them with matrix algebras of analytic functions in two variables. We look at the connections with semicrossed by -actions.     

Sets of all periodic points of a toral automorphism

In this article, subsets of T² that can arise as sets of all periodic points of a continuous 2-dimensional toral automorphism are characterized. Here the torus T² is viewed as [0,1)×[0,1) as a group under coordinatewise addition modulo 1.