Endomorphisms of symbolic algebraic varieties. (English) Zbl 0998.14001
Summary: The theorem of J. Ax [Ann. Math. (2) 88, 239-271 (1968; Zbl 0195.05701)] says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and is 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.
MSC:
| 14A10 | Varieties and morphisms |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 03C60 | Model-theoretic algebra |
| 14E05 | Rational and birational maps |
Keywords:
regular selfmapping of a complex algebraic variety; surjunctivity; proregular mappings; proalgebraic spaces; proalgebraic varieties; model theory; symbolic dynamicsCitations:
Zbl 0195.05701References:
| [1] | 1 J. Ax: The elementary theory of finite fields. Ann. Math. 88 (2), 239-271 (1968) · Zbl 0195.05701 · doi:10.2307/1970573 |
| [2] | 2 J. Ax: Injective endomorphismsq of varieties and schemes. Pac. J. Math. 31 (1), 1-7 (1969) · Zbl 0194.52001 · doi:10.2140/pjm.1969.31.1 |
| [3] | A. Bialyncki-Barula, M. Rosenlicht: Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc. 13 , 200-203 (1962) · Zbl 0107.14602 · doi:10.2307/2034464 |
| [4] | A. Borel: Injective endomorphisms of algebraic varieties. Arch. Math. (Basel) 20 , 531-537 (1969) · Zbl 0189.21402 · doi:10.1007/BF01899460 |
| [5] | T.G. Ceccherini-Silberstein, A. Machi, F. Scarabotti: Amenable groups and cellular automata. Preprint (1998) |
| [6] | I.M. Degtyarev: Multidimensional value distribution theory. In: Several com- plex variables III, G.M. Khenikin (ed.). Springer, Berlin, Heidelberg 1989 |
| [7] | W. Gottschalk: Some general dynamical notions. LNM 318, pp. 120-125, Springer, Berlin, Heidelberg 1973 · Zbl 0255.54035 |
| [8] | F. Greenleaf: Invariant means on topological groups Van Nostrand. Math. Stud. 16 , (1969) · Zbl 0174.19001 |
| [9] | AI |
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