Equilibrium states for interval maps: the potential $-t\log |Df|$

Bruin, Henk; Todd, Mike

Equilibrium states for interval maps: the potential

- t log | D f |

[États d’équilibre pour applications de l’intervalle : le potentiel

- t log | D f |

]

Bruin, Henk ; Todd, Mike

Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 559-600 / Harvested from Numdam

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Résumé
Abstract

Soit $f : I \to I$ une application multimodale de classe $C^{2}$ dont les dérivées le long des orbites des points critiques sont à croissance polynomiale, où $I$ est un intervalle. Nous démontrons l’existence et l’unicité d’un état d’équilibre pour le potentiel $φ_{t} : x \mapsto - t log | D f (x) |$ lorsque $t$ est proche de $1$ , et que la fonction de pression $t \mapsto P (φ_{t})$ est analytique sur un intervalle approprié près de $t = 1$ .

Let $f : I \to I$ be a $C^{2}$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $φ_{t} : x \mapsto - t log | D f (x) |$ for $t$ close to $1$ , and also that the pressure function $t \mapsto P (φ_{t})$ is analytic on an appropriate interval near $t = 1$ .

Publié le : 2009-01-01

MR 2568876 | Zbl 1192.37051 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2103
Classification: 37D35, 37D25, 37E05
Mots clés: États d'équilibre, formalisme thermodynamique, applications de l'intervalle, hyperbolicité non-uniforme

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     author = {Bruin, Henk and Todd, Mike},
     title = {Equilibrium states for interval maps: the potential $-t\log |Df|$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {42},
     year = {2009},
     pages = {559-600},
     doi = {10.24033/asens.2103},
     mrnumber = {2568876},
     zbl = {1192.37051},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2009_4_42_4_559_0}
}

Bruin, Henk; Todd, Mike. Equilibrium states for interval maps: the potential $-t\log |Df|$. Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009) pp. 559-600. doi : 10.24033/asens.2103. http://gdmltest.u-ga.fr/item/ASENS_2009_4_42_4_559_0/

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