## Ergodic Theory and Zd ActionsThe classical theory of dynamical systems has tended to concentrate on Z-actions or R-actions. In recent years, however, there has been considerable progress in the study of higher dimensional actions (i.e. Zd or Rd with d>1). This book represents the proceedings of the 1993-4 Warwick Symposium on Zd actions. It comprises a mixture of surveys and original articles that span many of the diverse facets of the subject, including important connections with statistical mechanics, number theory and algebra. |

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### Contents

Ergodic Ramsey Theoryan Update | 1 |

a review | 63 |

A Survey | 113 |

Boundaries Of Invariant Markov Operators The Identification Problem | 127 |

Squaring And Cubing The Circle RudolphS Theorem | 177 |

A Survey of Recent KTheoretic Invariants for Dynamical Systems | 185 |

Miles of Tile | 237 |

the size of a dynamically defined Cantorset | 259 |

A note on certain rigid subshifts | 307 |

On Representation of Integers in Linear Numeration Systems | 345 |

The structure of ergodic transformations conjugate to their inverses | 369 |

Approximation by periodic transformations and diophantine approximation of the spectrum | 387 |

Invariant algebras for σactions and their applications | 403 |

Large Deviations For Paths And Configurations Counting | 415 |

A Zeta Function For ZActions | 433 |

The Dynamical Theory Of Tilings And Quasicrystallography | 451 |

Uniformity in the Polynomial Szemerédi Theorem | 273 |

Entropy and Mixing | 297 |

Approximation Of Groups And Group Actions The Cayley Topology | 475 |

### Common terms and phrases

abelian action algebraic assume automorphism Borel C’-algebra compact metric space compactiﬁcation computable conﬁgurations conjecture consider convergence Corollary corresponding countable deﬁned deﬁnition denote dense density discrete dynamical systems equivalence relation ergodic theory example exists ﬁnite ﬁnite type ﬁrst ﬁxed ﬂows Furstenberg geodesic graph group G Hales-Jewett theorem harmonic functions Hausdorff dimension homeomorphism homogeneous spaces horospherical implies inﬁnite integer invariant measures IP-set isomorphic lattice in G Lebesgue Lemma Let G Lie group linear manifolds Markov operators Martin boundary Math measure preserving metric space minimal one-parameter subgroup orbit partition Penrose tilings periodic tilings Poisson boundary polynomial preprint probability measure proof prototiles proved Ramsey theory random walks recurrence result rotation satisﬁes sequence solvable groups speciﬁcation spectrum subset subshifts of ﬁnite symmetric spaces Theorem 2.1 topological transformations trivial ultraﬁlters unipotent uniquely ergodic vector weakly mixing