### Abstract

In this survey we shall present some relations between measure theory and geometric topology in dynamics. One of these relations comes as follows, on one hand from topological information of the system, some structure should be preserved by the dynamics at least in some weak sense, on the other hand, measure theory is soft enough that an invariant geometric structure almost always appears along some carefully chosen invariant measure. As an example, we have the known result that in dimension 2 the system has asymptotic growth of hyperbolic periodic orbits at least equal to the largest exponent of the action in homology.

Original language | English (US) |
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Title of host publication | Proceedings of the International Congress of Mathematicians 2010, ICM 2010 |

Pages | 1760-1776 |

Number of pages | 17 |

State | Published - Dec 1 2010 |

Event | International Congress of Mathematicians 2010, ICM 2010 - Hyderabad, India Duration: Aug 19 2010 → Aug 27 2010 |

### Publication series

Name | Proceedings of the International Congress of Mathematicians 2010, ICM 2010 |
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### Other

Other | International Congress of Mathematicians 2010, ICM 2010 |
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Country | India |

City | Hyderabad |

Period | 8/19/10 → 8/27/10 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Rodriguez Hertz, F. J. (2010). Measure theory and geometric topology in dynamics. In

*Proceedings of the International Congress of Mathematicians 2010, ICM 2010*(pp. 1760-1776). (Proceedings of the International Congress of Mathematicians 2010, ICM 2010).