### Abstract

Let G be a discrete group which admits an amenable action on a compact space and γ ∈ Aut(G) be an automorphism. We define a notion of entropy for y and denote the invariant by ha(γ). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(γ) = h_{Top}(γ̂) where h_{Top}(γ̂) denotes the topological entropy of the induced automorphism γ̂ of the (compact, abelian) dual group Ĝ. ha(·) enjoys a number of basic properties which are useful for calculations. For example, it decreases in invariant subgroups and certain quotients. These basic properties are used to compute the dual entropy of an arbitrary automorphism of a crystallographic group.

Original language | English (US) |
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Pages (from-to) | 711-728 |

Number of pages | 18 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 22 |

Issue number | 3 |

DOIs | |

State | Published - Aug 29 2002 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Ergodic Theory and Dynamical Systems*,

*22*(3), 711-728. https://doi.org/10.1017/S0143385702000366