Topological entropy and algebraic entropy for group endomorphisms.

*(English)*Zbl 1300.54002
Arhangel’skii, Alexander V. (ed.) et al., Proceedings of the international conference on topology and its applications (ICTA 2011), Islamabad, Pakistan, July 4–10, 2011. Cambridge: Cambridge Scientific Publishers (ISBN 978-1-908106-17-9/pbk). 133-214 (2012).

This is an extensive survey of the theory of entropy across several different fields of mathematics, with a primary emphasis on the topological and algebraic entropy of continuous endomorphisms of locally compact groups. The main inter-relationships between different notions of entropy in these settings are described and proved, and some new entropy-like functions are defined. Connections between the quantities and structures arising in these studies to other fields are described, including in particular Lehmer’s problem in algebraic number theory and Milnor’s problem in geometric group theory. An extensive bibliography covers both the history and sources for results that are not proved in the paper.

For the entire collection see [Zbl 1283.54001].

For the entire collection see [Zbl 1283.54001].

Reviewer: Thomas B. Ward (Durham)

##### MSC:

54-02 | Research exposition (monographs, survey articles) pertaining to general topology |

54H11 | Topological groups (topological aspects) |

22D05 | General properties and structure of locally compact groups |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

54C70 | Entropy in general topology |

28D20 | Entropy and other invariants |

37B40 | Topological entropy |

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

20F65 | Geometric group theory |

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\textit{D. Dikranjan} and \textit{A. Giordano Bruno}, in: Proceedings of the international conference on topology and its applications (ICTA 2011), Islamabad, Pakistan, July 4--10, 2011. Cambridge: Cambridge Scientific Publishers. 133--214 (2012; Zbl 1300.54002)

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