We extend the Ahlbrandt–Ziegler analysis of interpretability in א0-categorical structures by showing that existential interpretation is controlled by the monoid of self–embeddings and positive existential interpretation of structures without constant endomorphisms is controlled by the monoid of endomorphisms in the same way as general interpretability is controlled by the automorphism group.

We start with a modular K-system that possesses Anosov endomorphisms in both time directions. We extend the system via a crossed product construction. If one of the Anosov endomorphisms can be extended, then also the other one. Also the extensions form a Moebius group and allow a time reeection operator that implements on the algebra an antiisomorphism.

We introduce the universal functorial Lefschetz invariant for endomorphisms of finite CW -complexes in terms of Grothendieck groups of endomorphisms of finitely generated free modules. It encompasses invariants like Lefschetz number, its generalization to the Lefschetz invariant, Nielsen number and L2-torsion of mapping tori. We examine its behaviour under fibrations.

We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tyurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.

We study a family of H-to-1 conservative ergodic endomorphisms which we will show to admit no rj-finite absolutely continuous invariant measure. We exhibit recurrent measures for these transformations and study their ratio sets; the examples can be realized as C°° endomorphisms of the 2-torus.

The relative ranks of the monoid of endomorphisms of a strong independence algebra of infinite rank modulo two distinguished subsets are calculated. These subsets are the group of automorphisms and the endomorphisms α satisfying α2 = α. The extra generators are characterised in each case. 2000 Mathematics Subject Classification. 20M20, 08A35.

We construct two practical algorithms for twisted conjugacy classes of polycyclic groups. The first algorithm determines whether elements a group are conjugate given endomorphisms, under the condition that their Reidemeister coincidence number is finite. second representatives endomorphisms if finite, or returns “false” this infinite. also discuss theoretical extension these to polycyclic-by-fi...