Endomorphisms of symbolic algebraic varieties

The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory, algebraic geometry, and symbolic dynamics. 1. Ax’ surjunctivity theorem 1.A. Strict embeddings and surjunctivity. A map between sets is called a strict embedding, denoted f : X ⊂ 6= Y , if it is one-to-one but not onto. Then, following Gottschalk, (see [Gott, 1972]) a map f : X → Y is called surjunctive if it is not a strict embedding. In other words f is surjunctive iff it is either surjective or non-injective. 1.B. Theorem [Ax]1. Every regular selfmapping of a complex algebraic variety X is surjunctive. In other words no X admits a strict embedding X ⊂ 6= X. Or, put it yet another way, “one-to-one” implies “onto” for every regular map f : X → X. If X = Cn this specializes to the following earlier result by BialynickiBarula and Rosenlicht, (see [BB-R, 1962]). 1.B′. Every complex polynomial self-mapping of Cn is surjunctive. Repeat, this signifies that no strict polynomial embedding Cn ⊂ 6= Cn is possible, i.e. every injective polynomial map Cn → Cn is surjective. M. Gromov: IHES, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France, e-mail:[email protected] Mathematics Subject Classification (1991): 14A10, 14E10, 05C25, 03C45