## Dynamics of Foliations, Groups and PseudogroupsFoliations, groups and pseudogroups are objects which are closely related via the notion of holonomy. In the 1980s they became considered as general dynamical systems. This book deals with their dynamics. Since "dynamics” is a very extensive term, we focus on some of its aspects only. Roughly speaking, we concentrate on notions and results related to different ways of measuring complexity of the systems under consideration. More precisely, we deal with different types of growth, entropies and dimensions of limiting objects. Invented in the 1980s (by E. Ghys, R. Langevin and the author) geometric entropy of a foliation is the principal object of interest among all of them. Throughout the book, the reader will find a good number of inspirating problems related to the topics covered. |

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### Contents

Dynamical systems | 1 |

12 First examples | 3 |

13 Foliations laminations and holonomy | 6 |

14 Markov pseudogroups | 15 |

15 Hyperbolic spaces and groups | 20 |

Growth | 33 |

21 Growth types | 34 |

22 Growth in groups | 36 |

36 Entropy and resiliency | 89 |

Invariant measures | 97 |

42 Transverse invariant measures and homology | 101 |

43 Measures and orbit growth | 102 |

44 Transverse invariant measures in codimension 1 | 104 |

45 Vanishing entropy and invariant measures | 106 |

46 Entropy geodesic flow and invariant measures | 108 |

47 Harmonic measures | 125 |

23 Orbit growth for pseudogroups | 44 |

24 Expansion growth | 51 |

Entropy | 61 |

31 Entropy of classical systems | 62 |

312 Invariant measures | 65 |

313 Measuretheoretic entropy | 66 |

314 Examples | 68 |

315 Variational principle | 70 |

32 Entropy of pseudogroups | 73 |

33 Geometric entropy of foliations | 76 |

34 Relating various entropies | 79 |

35 Examples and constructions | 84 |

352 Gluing | 85 |

353 Turbulization | 86 |

48 PattersonSullivan measures | 142 |

Hausdorff dimension | 155 |

52 Julia sets | 161 |

53 Dimension in foliated manifolds | 166 |

54 Dimension of a hyperbolic boundary | 176 |

55 Dimension of a limit set | 179 |

Varia | 183 |

62 Expansive systems | 187 |

63 Pseudoorbits and pseudoleaves | 191 |

64 Generic leaves | 201 |

211 | |

223 | |