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## Local product structure for equilibrium states (2000)

Venue: | Trans. Amer. Math. Soc |

Citations: | 12 - 10 self |

### Citations

924 |
Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Second revised edition. With a preface by David Ruelle. Edited by Jean-Renau Chazottes
- Bowen
- 2008
(Show Context)
Citation Context ...be a compact Riemannian manifold, and let f : M →M be a C1+α mixing diffeomorphism, with the axiom-A property. We assume that a metric adapted to f has been choosen such that the following holds (see =-=[4]-=- for such metric) (1) Periodic points are dense in the set of non-wandering points Ω. (2) Ω is hyperbolic, meaning that for all x in Ω there is a continuous splitting of the tangent space TxM with (a)... |

249 |
Real and complex analysis. Third edition
- Rudin
- 1987
(Show Context)
Citation Context ... y∈Antn(x) rn(Cn(y))=k eSn(BS)(y,Cn(y)) < δλnS where Cn(y) denotes an n-cylinder containing y, and r n(Cn(y)) denotes the n-th return time. Fnδ = ∪rn(Cn(y))≤NCn(y) is compact. By Urysohn’s Lemma (see =-=[14]-=-), there exists some continuous function ϕδ,n, such that ϕδ,n(x) = 1, ,for all x ∈ Kn−1, 0 ≤ ϕδ,n ≤ 1, and ϕδ,n(x) = 0, ,for all x ∈ F n δ . Then µS(Kn) ≤ 1 λnS ∫ LnS(ϕδ,n)dµS ≤ δ, hence µS(H) = 0. 4.... |

73 |
On the fundamental ideas of measure theory.
- Rohlin
- 1949
(Show Context)
Citation Context ...uivalent on each W uloc(x) to the conditional measure µ A,u of the Equilibrium State µA with respect to any measurable partition subordinate to the unstable foliation. We refer the reader to [11] and =-=[13]-=- for precise definitions about measurable partitions and subordinate partitions. The proof of theorem A needs the absolute continuity of the conditional measures of the Equilibrium State. We shall fir... |

56 |
Some systems with unique equilibrium state
- Bowen
- 1974
(Show Context)
Citation Context ...s a unique Equilibrium State for BS, which is the unique gF - invariant quasi-Gibbs measure associated to BS. Moreover, the pressure of the system is log λS. Proof. Using ideas given in [15], [9] and =-=[3]-=-, we check that we have just to prove the following lemma. Lemma 5.10. There exists a universal constant C such that, for all n ∈ N, for all n-cylinders Cn, andforallx in Cn we have 1 C mS(Cn) ≤ ≤ C. ... |

35 |
Unique ergodicity for horocycle foliations
- BOWEN, MARCUS
- 1977
(Show Context)
Citation Context ...other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven. 1. Introduction In =-=[5]-=-, Bowen and Marcus define a notion of the transversal to a n-dimensional foliation G in a metric space X and a notion of a G-invariant measure. They prove that up to a constant there is a unique G-inv... |

27 |
The metric entropy on diffeomorphisms: Part I: Characterization of measures satisfying Pesin’s entropy formula
- Ledrappier, Young
- 1985
(Show Context)
Citation Context ...tem is equivalent on each W uloc(x) to the conditional measure µ A,u of the Equilibrium State µA with respect to any measurable partition subordinate to the unstable foliation. We refer the reader to =-=[11]-=- and [13] for precise definitions about measurable partitions and subordinate partitions. The proof of theorem A needs the absolute continuity of the conditional measures of the Equilibrium State. We ... |

26 |
Theorie ergodique pour des classes d’operations non completementaires continues,
- T, Marinescu
- 1950
(Show Context)
Citation Context ...atifies the required properties. Appendix C. Ionescu-Tulcea and Marinescu’s theorem We recall here Ionescu-Tulcea and Marinescu’s theorem, and some useful lemmas. Some complete proofs can be found in =-=[10]-=- or [6]. Theorem C.1 (Ionescu-Tulcea & Marinescu). Let (V, ‖ ‖V) and (U , ‖ ‖U ) be two C -Banach spaces such that V ⊂ U . We assume that (i) if (ϕn)n∈IN is a sequence of functions in V which conver... |

25 |
Geodesic paths and horocycle flow on abelian covers. Lie groups and ergodic theory
- Babillot, Ledrappier
- 1996
(Show Context)
Citation Context ...s ◦ ρx,x ′ − F ◦ Φs) ds where F is any Höldercontinuous function Ω → R. In his proof, Haydn assumes that Φs∗µx and µ Φs(x) are two equivalent measures and uses the symbolic dynamic just as in [5]. In =-=[1]-=-, Babillot and Ledrappier prove the previous result, plus uniqueness, without assuming that Φs∗µx and µ Φs(x) are absolutely continuous. Unfortunately, they still use the symbolic dynamic, which preve... |

17 |
Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification.
- Haydn, Ruelle
- 1992
(Show Context)
Citation Context ...ere exists a unique Equilibrium State for BS, which is the unique gF -invariant quasiGibbs measure associated to BS . Moreover, the pressure of the system is log λS. Proof. Using ideas given in [15], =-=[9]-=- and [3], we check that we have just to prove the following lemma. Lemma 5.10. There exists a universal constant C such that for all n ∈ IN, for all ncylinder Cn, and for all x in Cn we have 1 C ≤ mS(... |

17 |
The metric entropy of diffeomorphisms. Part II: relations between entropy, exponents and dimension
- Ledrappier, Young
- 1985
(Show Context)
Citation Context ... any x and y in Γ, µA,ux and µ A,u y are equivalent modulo holonomy, hence µ A,u x is equivalent to mA for every x in F . This proves that µ A has a local product structure in R and then in all Ω. By =-=[12]-=-, we know the existence of a set ∆ of full µA-measure of points such that µA,sx and µA,ux have pointwise dimensions equal δs and δu. Pick any point x in ∆ ∩ Γ. Because of the continuity of x 7→ Eu(x) ... |

15 |
Thermodynamic formalism for maps satisfying positive expansiveness and specification.
- Ruelle
- 1992
(Show Context)
Citation Context ... SS there exists a unique Equilibrium State for BS, which is the unique gF -invariant quasiGibbs measure associated to BS . Moreover, the pressure of the system is log λS. Proof. Using ideas given in =-=[15]-=-, [9] and [3], we check that we have just to prove the following lemma. Lemma 5.10. There exists a universal constant C such that for all n ∈ IN, for all ncylinder Cn, and for all x in Cn we have 1 C ... |

10 | Dimension of hyperbolic measures – a proof of the Eckmann-Ruelle Conjecture, preprint
- Barreira, Pesin, et al.
(Show Context)
Citation Context ...re constant, and δ = δ u + δ s .s1892 RENAUD LEPLAIDEUR Remark. This last equality δ = δ u + δ s is a particular case of a general fact in the non-uniform hyperbolic case that has just been proved in =-=[2]-=-. 3. The dynamical system (F, gF ) 3.1. Definitions. Pick R one particular proper rectangle with the Markov property. By Poincaré’s theorem, we define respectively Rn and R∞ as the set of points in R ... |

7 |
Finite and σ-finite invariant measures
- Dowker
- 1951
(Show Context)
Citation Context ...sts a Φ-invariant ergodic Borel probability measure such that (X,Φ, µ) induces by the first return map the system (Y,Ψ, ν). Proof. We give here just the sketch of the proof. For a complete proof, see =-=[7]-=-. Denote by l def = 1R r dν . Pick any Borel set in X, and any integer k. Set Ak def = {y ∈ A such that Φ−k(y) ∈ Y and Φ−j(y) /∈ Y ∀ j < k} and define µ(A) def = l × ( +∞∑ k=0 ν ( Φ−k(Ak) )) . Just ch... |

5 |
Canonical product structure of equilibrium states Random Comput
- Haydn
- 1994
(Show Context)
Citation Context ...variant measure. They prove that up to a constant there is a unique G-invariant measure for any Axiom-A diffeomorphism or flow where G is the strong stable or unstable foliation for any basic set. In =-=[8]-=-, for the AxiomA flow case (Ω,Φ), Haydn proves the existence of a transversal system of measure {µx} supported on the local weak unstable manifold which is not invariant along the W ssfoliation but wh... |

1 |
Méthode de opérateurs de transferts : Transformations dilatantes de l’intervalle et dénombrement de géodésiques fermées
- Broise, Dal’bo, et al.
(Show Context)
Citation Context ...the required properties. Appendix C. Ionescu-Tulcea and Marinescu’s theorem We recall here Ionescu-Tulcea and Marinescu’s theorem, and some useful lemmas. Some complete proofs can be found in [10] or =-=[6]-=-. Theorem C.1 (Ionescu-Tulcea & Marinescu). Let (V, ‖ ‖V) and (U , ‖ ‖U ) be two C -Banach spaces such that V ⊂ U . We assume that (i) if (ϕn)n∈IN is a sequence of functions in V which converges in ... |