Wolfgang Krieger

A class ofC*-algebras and topological Markov chains

In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a matrix A =(A (i,j))i,~ z, Z a finite set, A(i,j)c{0, l}, and where every row and every column of A is non-zero. (That A(i,j)e{O, 1} is assumed for convenience only. All constructions and results extend to matrices with entries in 2~+. We comment on this in Remark 2.18.) A C*-algebra 6~ A is then generated by partial isometries Si~O(i~X ) that act on a Hilbert space in such a way that their support projections Qi=S*S~ and their range projections P~ =SIS* satisfy the relations

On entropy and generators of measure-preserving transformations

Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy h(T). We prove that T has a generator of size k where eh(T) _ k < eh(T)+ 1

On dimension functions and topological Markov chains

Let X be a second countable zero-dimensional a-compact space. Denote by ~ x the Boolean ring of compact open subsets of X and consider a countable locally finite group N of homeomorphisms of X. Assume that every element of G has an open fixed point set, and, for the moment, assume also, that the fixed point set of every element of ff has a compact complement. If we are given a finite family A~, ia~, of disjoint compact open non-empty subsets of X together with a family g~(-f, i e J , such that the g~A~, i ~ J , are also disjoint, and such that

On the uniqueness of the equilibrium state

More generally, we consider the subshifts of F z. These are the closed subsets of F z that are invariant under S. We denote for such a subshift X by C(X) the set of continuous real-valued functions on X, and by d//(X) the set of shiftinvariant Borel probability measures on X. h(/z) will stand for the entropy of a tz E rig(X). (For the entropy and its properties see, e.g., [11].) We set for a subshift X,

On sofic systems. I

The results ofOn sofic systems I on topological Markov chains extending sofic systems are completed. To homomorphisms of sofic systems are canonically associated homomorphisms of Markov extensions. Also considered is a class of finitary codes for sofic systems.

On the subsystems of topological Markov chains

Let S A be an irreducible and aperiodic topological Markov chain. If S Ā is an irreducible and aperiodic topological Markov chain, whose topological entropy is less than that of S A , then there exists an irreducible and aperiodic topological Markov chain, whose topological entropy equals the topological entropy at S Ā , and that is a subsystem of S A . If S is an expansive homeomorphism of the Cantor discontinuum, whose topological entropy is less than that of S A , and such that for every j ∈ℕ the number of periodic points of least period j of S is less than or equal to the number of periodic points of least period j of S A , then S is topological conjugate to a subsystem of S A .

On the finitary isomorphisms of markov shifts that have finite expected coding time

SummaryA necessary condition is given for a finitary isomorphism between mixing Markov shifts of equal entropy to have finite expected coding time.

On Subshifts and Semigroups

Semigroups (with zero) and inverse semigroups (with zero) are constructed from subshifts. The semigroups and inverse semigroups that are obtained by this construction are characterized and classified.

On a class of hyperfinite factors that arise from null-recurrent Markov chains

Abstract Let α > 1, and let p α be the probability measure on {0, 1} that is given by p α {0} p α {1} −1 = α . The group of non-singular transformations of Π n = 1 ∞ ({0, 1}, P {0, 1}, p α ) that is generated by {S k x:κ ϵ N }, S k x = ( x n + δ kn (mod 2)) n =1 ∞ , gives rise, by the group measure space construction, to a hyperfinite factor Ol α . Using null-recurrent Markov chains we construct a family Ol αq , 1 2 ⩽ q , of mutually non-isomorphic hyperfinite factors such that Ol αq ⊗ Ol αq is isomorphic to Ol α , 1 2 ⩽ q .

On Quasi-Invariant Measures in Uniquely Ergodic Systems

Let X be a compact metrizable space, ~ (X) its G-algebra of Borel sets, and let T be a homeomorphism of X. We denote for a Borel measure # on X by T/2 the measure that is obtained by setting TI~(A)=I~(T -1 A), A e ~ . There exists always a Borel probability measure # such that T/2=# [10, p. 76]. If there exists only one such invariant measure then T is called uniquely ergodic. A measure # is called quasi-invariant for T if T# is equivalent to/2. In this paper we show that every uniquely ergodic homeomorphism of a compact metrizable space whose invariant measure is non-atomic possesses more than countably many non-atomic quasi-invariant ergodic measure classes. The problem of describing such measure classes has arisen in the theory of unitary representations of locally compact groups [12, p. 650]. In fact, we shall prove a more precise result. To state this result we need more terminology. Let (Y, ~ ) be a Borel structure and let ~ be an equivalence relation in Y. It can happen that the G-algebra ~ contains sets A i, i eN , such that