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...ing. For each such value a, we have the corresponding values αa and βa. We point out, however, that it also has meaning to consider real values of a (any positive real is possible) in several of our results (which are not related to the renormalization operator for the shift). 5. Proof of Theorem C The proof has 3 main steps. In the first subsection we give an important result on the control of the variation of the potential on cylinders. In the second subsection we recall the construction of Gibbs states obtained by induction on a cylinder. We recall and use the method that was introduced in [19]. In particular, we introduce a one-family parameter of Transfer Operator, introduced the critical parameter Sc and we show that existence of Gibbs state is dependent of the fact that the operator has spectral radius larger than 1 or not close to the critical value Sc. In the last section we study the realization of this condition for our family of potentials in function of the parameters. 5.1. Distortion on cylinders. We recall that the potential φβ is defined as follows: φβ(x) = −2 log ( θβ ◦ σ(x) θβ(x) ) if x ∈ [0], φ(x)β = −2 log ( 2β − 1− θβ ◦ σ(x) 2β − 1− θβ(x) ) if x ∈ [1]. The theory o...

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... HB and JRC for readings and advises. 12 RENAUD LEPLAIDEUR The goal of this note is to present a method based on induction to answer to these two previous questions. Thismethod wasactuallypresentedin=-=[9]-=-forthecaseofuniformlyhyperbolicdynamical system and Hölder continuous potential. It was then developed and extended in further works of the author; the note summarizes here all the principal steps usi...

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...on f is Gâteaux-differentiable at x in the direction y if exists. f(x + ty) − f(x) lim t→0 t 1.4. Overview of the paper. The main tool is the notion of local equilibrium state as it was introduced in =-=[17]-=- and developed in further works of the author. We briefly recall in Appendix A the principal steps.NON-FLAT PHASE TRANSITIONS 5 In Section 2 we prove main of the results concerning Theorem A and the ...

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...nterval. In section 6 we give a dynamical proof of the first statement of the theorem. 3. Large deviations for return time in cylinders We first recall the local thermodynamic formalism introduced in =-=[4]-=-. Then we recall how the large deviations principle for union of cylinders was obtained in [2]. Finally, we derive a uniform mass concentration principle. 3.1. Induced systems and local thermodynamic ...

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...441612, version 1 - 16 Dec 2009 The proof here is somehow different. The key point is that there are local product structures, both for coordinates (see e.g. [Bow75]) and for Gibbs measures (see e.g. =-=[Lep00]-=-). Moreover, if we locally set µϕ ≈ µ s ϕ ⊗ µu ϕ , also satisfy some Gibbs property. these two measures µ u ϕ and µsϕ Using these local coordinates, a ball B((x,y),ε) can be approximate by a cylinder ...

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...me one-parameter family of transfer operators. These transfer operators act on the induced system on an unstable leaf of reference crossing the set A upon consideration. This construction was done in =-=[17]-=- for other purposes than studying return times (namely to construct equilibrium states). Once we have this free energy for Poincaré cycles and its properties, we apply two results (that are little use...

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...val. In section 6 we give a dynamical proof of the statement in point 1 if the theorem. 3. Large deviation for return time in cylinders We first recall the local thermodynamic formalism introduced in =-=[4]-=-. Then we recall how the large deviation principle for union of cylinders was obtained in [2]. Finally, we derive a uniform mass concentration principle. 3.1. Induced systems and local thermodynamic f...

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...tentials. It is now a classical way to consider induction for studying the thermodynamic formalism for non-uniformly hyperbolic dynamical systems. Several relatively different methods exist (see e.g. =-=[12, 19, 3, 20]-=-). But for all them, the main problem is that they may exist measures which are not seen by the induction; therefore, it is always needed to check if the maximum obtained is the global one. In other w...

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... state of ϕ. If ϕ = ϕu(g) then µϕy are absolutely continuous measures induced by the Riemannian metric a. For more details about the system {µϕy} and the proof of existence and uniqueness, see, e.g., =-=[L00]-=-. A.4. Sketch of the proof of Lemma 8. Let ξ(x0) = idT2 for some x0 ∈ ∂D k. Then structural stability gives the continuation ξ : Dk → Top0(T 2) to the rest of the disk of the conjugacy to the linear m...