Algebraic entropy for Abelian groups.

*(English)*Zbl 1176.20057The notion of entropy associated to some mathematical structures (maps) was first introduced by R. L. Adler, A. G. Konheim, M. H. McAndrew [Trans. Am. Math. Soc. 114, 309-319 (1965; Zbl 0127.13102)]. In this paper the authors note that it “can be tailored to fit mappings on other mathematical structures”, and they propose a notion of entropy associated to endomorphisms of Abelian groups. In the present paper this kind of entropy, called ‘algebraic entropy’, is studied.

Basic definitions and properties are listed in Section 1. It is proved that every infinite direct sum of non-zero Abelian \(p\)-groups admits an endomorphism (a shift map) of infinite algebraic entropy (Theorem 1.12). Moreover, if \(G\) is a reduced \(p\)-group which has an endomorphism of finite and strictly positive entropy then \(G\) has an infinite bounded summand (Theorem 1.19). Connections with ergodic theory are exhibited in Section 2. It is proved that an endomorphism of an Abelian \(p\)-group is recurrent if and only if it has zero entropy (Proposition 2.9). Using this result the authors observe that there exists a separable \(p\)-group which admits recurrent endomorphisms which are not strongly recurrent (Proposition 2.12).

A very important result, called the Addition Theorem, is proved in Section 3. It states that, for torsion groups, the entropy is additive with respect invariant subgroups (Theorem 3.1).

The rest of the paper is dedicated to study some important particular cases. It is proved that: there exists a standard essentially indecomposable group \(G\) which admits an endomorphism of infinite entropy (Theorem 4.4); every small endomorphism of a semi-standard \(p\)-group has zero entropy (Theorem 5.2). In the end of the paper it is proved that the algebraic entropy is the unique non-negative, real numerical invariant associated to the endomorphisms, which satisfies certain properties (Theorem 6.1).

Basic definitions and properties are listed in Section 1. It is proved that every infinite direct sum of non-zero Abelian \(p\)-groups admits an endomorphism (a shift map) of infinite algebraic entropy (Theorem 1.12). Moreover, if \(G\) is a reduced \(p\)-group which has an endomorphism of finite and strictly positive entropy then \(G\) has an infinite bounded summand (Theorem 1.19). Connections with ergodic theory are exhibited in Section 2. It is proved that an endomorphism of an Abelian \(p\)-group is recurrent if and only if it has zero entropy (Proposition 2.9). Using this result the authors observe that there exists a separable \(p\)-group which admits recurrent endomorphisms which are not strongly recurrent (Proposition 2.12).

A very important result, called the Addition Theorem, is proved in Section 3. It states that, for torsion groups, the entropy is additive with respect invariant subgroups (Theorem 3.1).

The rest of the paper is dedicated to study some important particular cases. It is proved that: there exists a standard essentially indecomposable group \(G\) which admits an endomorphism of infinite entropy (Theorem 4.4); every small endomorphism of a semi-standard \(p\)-group has zero entropy (Theorem 5.2). In the end of the paper it is proved that the algebraic entropy is the unique non-negative, real numerical invariant associated to the endomorphisms, which satisfies certain properties (Theorem 6.1).

Reviewer: Simion Sorin Breaz (Cluj-Napoca)

##### MSC:

20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |

20K10 | Torsion groups, primary groups and generalized primary groups |

37A35 | Entropy and other invariants, isomorphism, classification in ergodic theory |

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\textit{D. Dikranjan} et al., Trans. Am. Math. Soc. 361, No. 7, 3401--3434 (2009; Zbl 1176.20057)

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##### References:

[1] | R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309 – 319. · Zbl 0127.13102 |

[2] | D. Alcaraz, D. Dikranjan, M. Sanchis, Infinitude of Bowen’s entropy for group endomorphisms, preprint. · Zbl 1317.22001 |

[3] | Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401 – 414. · Zbl 0212.29201 |

[4] | A. L. S. Corner, On endomorphism rings of primary abelian groups, Quart. J. Math. Oxford Ser. (2) 20 (1969), 277 – 296. · Zbl 0205.32506 |

[5] | A. L. S. Corner, On endomorphism rings of primary abelian groups. II, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 5 – 13. · Zbl 0326.20047 |

[6] | A. L. S. Corner and Rüdiger Göbel, Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc. (3) 50 (1985), no. 3, 447 – 479. · Zbl 0562.20030 |

[7] | László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. · Zbl 0209.05503 |

[8] | L. Fuchs, Vector spaces with valuations, J. Algebra 35 (1975), 23 – 38. · Zbl 0318.15002 |

[9] | L. Fuchs and J. M. Irwin, On \?^{\?+1}-projective \?-groups, Proc. London Math. Soc. (3) 30 (1975), part 4, 459 – 470. · Zbl 0324.20059 |

[10] | László Fuchs and Luigi Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001. · Zbl 0973.13001 |

[11] | P. D. Hill and C. K. Megibben, Quasi-closed primary groups, Acta Math. Acad. Sci. Hungar 16 (1965), 271 – 274 (English, with Russian summary). · Zbl 0209.33204 |

[12] | Charles Megibben, Large subgroups and small homomorphisms, Michigan Math. J. 13 (1966), 153 – 160. · Zbl 0166.02502 |

[13] | Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. · Zbl 0123.03402 |

[14] | Justin Peters, Entropy on discrete abelian groups, Adv. in Math. 33 (1979), no. 1, 1 – 13. · Zbl 0421.28019 |

[15] | Justin Peters, Entropy of automorphisms on L.C.A. groups, Pacific J. Math. 96 (1981), no. 2, 475 – 488. · Zbl 0478.28010 |

[16] | Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. · Zbl 0507.28010 |

[17] | R. S. Pierce, Homomorphisms of primary abelian groups, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman and Co., Chicago, Ill., 1963, pp. 215 – 310. |

[18] | L. Salce, Struttura dei p-gruppi abeliani, Pitagora Ed., Bologna, 1980. · Zbl 0599.20087 |

[19] | 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). · Zbl 0648.22002 |

[20] | Michael D. Weiss, Algebraic and other entropies of group endomorphisms, Math. Systems Theory 8 (1974/75), no. 3, 243 – 248. · Zbl 0298.28014 |

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