Algebraic entropy for Abelian groups
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- by Dikran Dikranjan, Brendan Goldsmith, Luigi Salce and Paolo Zanardo PDF
- Trans. Amer. Math. Soc. 361 (2009), 3401-3434 Request permission
Abstract:
The theory of endomorphism rings of algebraic structures allows, in a natural way, a systematic approach based on the notion of entropy borrowed from dynamical systems. Here we study the algebraic entropy of the endomorphisms of Abelian groups, introduced in 1965 by Adler, Konheim and McAndrew. The so-called Addition Theorem is proved; this expresses the algebraic entropy of an endomorphism $\phi$ of a torsion group as the sum of the algebraic entropies of the restriction to a $\phi$-invariant subgroup and of the endomorphism induced on the quotient group. Particular attention is paid to endomorphisms with zero algebraic entropy as well as to groups all of whose endomorphisms have zero algebraic entropy. The significance of this class arises from the fact that any group not in this class can be shown to have endomorphisms of infinite algebraic entropy, and we also investigate such groups. A uniqueness theorem for the algebraic entropy of endomorphisms of torsion Abelian groups is proved.References
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
- D. Alcaraz, D. Dikranjan, M. Sanchis, Infinitude of Bowen’s entropy for group endomorphisms, preprint.
- Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414. MR 274707, DOI 10.1090/S0002-9947-1971-0274707-X
- A. L. S. Corner, On endomorphism rings of primary abelian groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 277–296. MR 258949, DOI 10.1093/qmath/20.1.277
- A. L. S. Corner, On endomorphism rings of primary abelian groups. II, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 5–13. MR 422453, DOI 10.1093/qmath/27.1.5
- A. L. S. Corner and Rüdiger Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), no. 3, 447–479. MR 779399, DOI 10.1112/plms/s3-50.3.447
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- L. Fuchs, Vector spaces with valuations, J. Algebra 35 (1975), 23–38. MR 371995, DOI 10.1016/0021-8693(75)90033-2
- L. Fuchs and J. M. Irwin, On $p^{\omega +1}$-projective $p$-groups, Proc. London Math. Soc. (3) 30 (1975), part 4, 459–470. MR 374288, DOI 10.1112/plms/s3-30.4.459
- László Fuchs and Luigi Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. MR 1794715, DOI 10.1090/surv/084
- P. D. Hill and C. K. Megibben, Quasi-closed primary groups, Acta Math. Acad. Sci. Hungar. 16 (1965), 271–274 (English, with Russian summary). MR 191957, DOI 10.1007/BF01904835
- Charles Megibben, Large subgroups and small homomorphisms, Michigan Math. J. 13 (1966), 153–160. MR 195939
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Justin Peters, Entropy on discrete abelian groups, Adv. in Math. 33 (1979), no. 1, 1–13. MR 540634, DOI 10.1016/S0001-8708(79)80007-9
- Justin Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981), no. 2, 475–488. MR 637984
- Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728
- R. S. Pierce, Homomorphisms of primary abelian groups, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman & Co., Chicago, Ill., 1963, pp. 215–310. MR 0177035
- L. Salce, Struttura dei p-gruppi abeliani, Pitagora Ed., Bologna, 1980.
- Luchezar N. Stojanov, Uniqueness of topological entropy for endomorphisms on compact groups, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 3, 829–847 (English, with Italian summary). MR 916296
- Michael D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75), no. 3, 243–248. MR 385834, DOI 10.1007/BF01762672
Additional Information
- Dikran Dikranjan
- Affiliation: Dipartimento di Matematica e Informatica, Università di Udine, Via Delle Scienze 206, 33100 Udine, Italy
- Email: dikranja@dimi.uniud.it
- Brendan Goldsmith
- Affiliation: School of Mathematical Sciences, Dublin Institute of Technology, Dublin 2, Ireland
- Email: brendan.goldsmith@dit.ie
- Luigi Salce
- Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
- MR Author ID: 153345
- Email: salce@math.unipd.it
- Paolo Zanardo
- Affiliation: Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy
- Email: pzanardo@math.unipd.it
- Received by editor(s): May 12, 2006
- Published electronically: March 3, 2009
- Additional Notes: The research of the first, third, and fourth authors was supported by MIUR, PRIN 2005.
- © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 3401-3434
- MSC (2000): Primary 20K30; Secondary 20K10, 37A35
- DOI: https://doi.org/10.1090/S0002-9947-09-04843-0
- MathSciNet review: 2491886
Dedicated: In Memoriam: Il Maestro, Adalberto Orsatti