Statistical properties of unimodal maps

Artur Avila; Carlos Gustavo Moreira

Publications Mathématiques de l'IHÉS (2005)

  • Volume: 101, page 1-67
  • ISSN: 0073-8301

Abstract

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We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.

How to cite

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Avila, Artur, and Moreira, Carlos Gustavo. "Statistical properties of unimodal maps." Publications Mathématiques de l'IHÉS 101 (2005): 1-67. <http://eudml.org/doc/104209>.

@article{Avila2005,
abstract = {We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.},
author = {Avila, Artur, Moreira, Carlos Gustavo},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {typical analytic unimodal maps; chaotic attractors; Lyapunov exponents; periodic orbits; critical orbit; physical measures},
language = {eng},
pages = {1-67},
publisher = {Springer},
title = {Statistical properties of unimodal maps},
url = {http://eudml.org/doc/104209},
volume = {101},
year = {2005},
}

TY - JOUR
AU - Avila, Artur
AU - Moreira, Carlos Gustavo
TI - Statistical properties of unimodal maps
JO - Publications Mathématiques de l'IHÉS
PY - 2005
PB - Springer
VL - 101
SP - 1
EP - 67
AB - We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.
LA - eng
KW - typical analytic unimodal maps; chaotic attractors; Lyapunov exponents; periodic orbits; critical orbit; physical measures
UR - http://eudml.org/doc/104209
ER -

References

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