Journal of the European Mathematical Society

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Volume 1, Issue 2, 1999, pp. 109–197
DOI: 10.1007/PL00011162

Endomorphisms of symbolic algebraic varieties

Misha Gromov[1]

(1) Institut des Hautes √Čtudes Scientifiques, 35, route de Chartres, Le Bois-Marie, 91440, Bures-sur-Yvette, France

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.

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Gromov Misha: Endomorphisms of symbolic algebraic varieties. J. Eur. Math. Soc. 1 (1999), 109-197. doi: 10.1007/PL00011162