The entropy conjecture for tori has been settled a long time ago by Misiurewicz and Przytycki. We show that their result extends from tori to compact and connected abelian groups. We give stronger results for hyperbolic automorphisms. Throughout this paper G denotes a compact and connected metric abelian group. The group operation on G is written additively.

We prove that the outer automorphism group Out(G) is residually finite when the group G is virtually compact special (in the sense of Haglund and Wise) or when G is isomorphic to the fundamental group of some compact 3-manifold. To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism φ of a group G is said to be commensurating, if...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p, then it also extends to the simple Cole extension with respect to p. W...

We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. Our main technical tools are the structural analysis of type III factors and the theory of canonical extension of endomorphisms introduced by Izumi.

Smooth compact complex surfaces admitting non-trivial surjective endomorphisms are classified up to isomorphisms. The algebraic case has been classified in [3], [19]. The following surfaces are listed in the non-algebraic case: a complex torus, a Kodaira surface, a Hopf surface with at least two curves, an Inoue surface with curves, and an Inoue surface without curves satisfying a rationality c...

Noncommutative topological entropy estimates are obtained for polynomial gauge invariant endomorphisms of Cuntz algebras, generalising known results for the canonical shift endomorphisms. Exact values for the entropy are computed for a class of permutative endomorphisms related to branching function systems introduced and studied by Bratteli, Jorgensen and Kawamura. In [Vo] D.Voiculescu defined...