Toshihiro Hamachi

On a Bernoulli shift with non-identical factor measures

There exists a Bernoulli shift with non-identical factor measures for which no invariant σ-finite equivalent measure exists.

Orbital factor map

For a certain pair of discrete measured ergodic equivalence relations, we specify how one is imbedded into the other and obtain a factor and its subfactor. For this pair of factors index theory is developed. We obtain an index formula, an ‘extended’ relation corresponding to the basic extension, and so on. Our formulas as well as construction are very explicit and are given in terms of ‘measure theoretic’ data. We also present examples showing the contrast between type III 0 index theory and type II 1 (or III λ ) index theory.

Nonsingular dynamical systems, Bratteli diagrams and Markov odometers

We prove that any ergodic non-singular transformation is orbit equivalent to a Markov odometer which is uniquely ergodic.

Subsystems of finite type and semigroup invariants of subshifts

Abstract Hamachi and Inoue obtained a necessary and sufficient condition for the embeddability of an irreducible subshift of finite type into a Dyck shift (Embedding of shifts of finite type into the Dyck shift, Monatsh. Math. 145 (2005), 107–129). Krieger introduced a property A of subshifts that is an invariant of topological conjugacy and he constructed for property A subshifts an invariantly associated semigroup (with zero) (On a syntactically defined invariant of symbolic dyanamics, Ergod. Th. Dynam. Sys. 20 (2000), 501–516). We introduce a class of property A subshifts, of which the Dyck shifts are prototypes, to which there are associated inverse semigroups (with zero) that arise from finite directed graphs. These subshifts also allow a suitable presentation by a finite directed graph that is labeled by elements of the associated inverse semigroup. We extend the criterion for embeddability of an irreducible subshift of finite type from the Dyck shifts to target shifts in this class.

Markov odometer actions not of product type

We find an explicit example of a Markov odometer action which is not orbit-equivalent to any product odometer action. We make use of some techniques from Kriegers construction of an action of non-product type arising from a flow. However, his example is not realized as an odometer action with an explicit measure, nor does it arise from a Markov measure.

On certain subshifts and their associated monoids

Within a subclass of monoids (with zero) a structural characterization is given of those that are associated to topologically transitive subshifts with Property (A).

Associated Actions and Uniqueness of Cocycles

We use the methods of orbital ergodic theory to show the existence of many strange cocycles. Any conservative ergodic flow is the associated action for some recurrent cocycle of an ergodic probability preserving transformation, and this cocycle is determined uniquely up to cohomology via orbit equivalence.

Isomorphic actions of group extensions on a measure space

Abstract Let E = E ( G , A ) be a group extension of an abelian 1.c.s.c. group A by an amenable 1.c.s.c. group G . An ergodic action V of A is said to be extendible to an action W of E if V(A) is isomorphic to the restriction of W onto the subgroup A ∋ E . The extension property is described and studied in terms of cocycles over a skew product with values in A . Several examples of R -actions are considered. We answer the question of when two isomorphic actions of A can be extended to isomorphic actions of E(G, A) .