On the commuting polynomial endomorphisms of C 2 We determine all couples of commuting polynomial endomorphisms of C 2 that extends to holomorphic endomorphisms of P 2 and that have disjoint sequences of iterates.

If M = Z, and G is a finite (nonabelian) group, then G is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation F : G − ←⊃ which commutes with all shift maps, and where nearby coordinates are combined together using the multiplication operator of G. We provide equivalent conditions for MCA to be group endomorphisms of G, and show how MCA on G inherit a natura...

In this article we discuss the interaction between the geometry of a quaternionKähler manifold M and that of the Grassmannian G3(g) of oriented 3-dimensional subspaces of a compact Lie algebra g. This interplay is described mainly through the moment mapping induced by the action of a group G of quaternionic isometries on M . We give an alternative expression for the endomorphisms I1, I2, I3, bo...

For a compact surface Σ (orientable or not, and with boundary or not) we show that the fixed subgroup, FixB, of any family B of endomorphisms of π1(Σ) is compressed in π1(Σ), i.e., rk(FixB) 6 rk(H) for any subgroup FixB 6 H 6 π1(Σ). On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, G, of finitely...

If M = ZD, and B is a finite (nonabelian) group, then B is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation G : B−→ B which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of B. We characterize when MCA are group endomorphisms of B, and show that MCA on B inherit a natural structure theory from t...

The unital endomorphisms of B(H) of (Powers) index n are classified by certain U(n)-orbits in the set of non-degenerate representations of the Cuntz algebra On on H. Using this, the corresponding conjugacy classes are identified, and a set of labels is given. This set of labels is P/ ∼ where P is a set of pure states on the UHFalgebra Mn∞ , and ∼ is a non-smooth equivalence on P . Several subse...

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K0 of the category of “φ-endomorphisms of finitely generated free RΠ(G,X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.

We introduce the notions of approximate innerness and central triviality for endomorphisms on separable von Neumann factors, and we characterize them for hyperfinite factors by Connes-Takesaki modules of endomorphisms and modular endomorphisms which are introduced by Izumi. Our result is a generalization of the corresponding result obtained by KawahigashiSutherland-Takesaki in automorphism case.

Étale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly étale rational endomorphisms of weak Calabi-Yau varieties.