2011/06/27 by A. T. Baraviera, Alexandre Baraviera, A. O. Lopes +6
Mathematics · Physics and Astronomy · #37A05 #37A60 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Probability (math.PR) #Quantum chaos and dynamical systems #Statistical Mechanics (cond-mat.stat-mech) #Theoretical and Computational Physics #cond-mat.stat-mech #math.DS #math.PR #msc:37A05 #msc:37A60
paper · pdf · doi:10.48550/arxiv.1106.5379
arxiv created 2011/06/27 · openalex publication_date 2011/06/27 · arxiv updated 2011/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose σ is the shift acting on Bernoulli space X=\0,1\^ℕ, and, consider a fixed function f:X → ℝ, under the Waters's conditions (defined in a paper in ETDS 2007). For each real value t≥ 0 we consider the Ruelle Operator Ltf. We are interested in the main eigenfunction ht of Ltf, and, the main eigenmeasure νt, for the dual operator Ltf^*, which we consider normalized in such way ht(0^∞)=1, and, ∫ ht d νt=1, ∀ t>0. We denote μt= ht νt the Gibbs state for the potential t f. By selection of a subaction V, when the temperature goes to zero (or, t→ ∞), we mean the existence of the limit V:=limt→∞(1)/(t)log(ht). By selection of a measure μ, when the temperature goes to zero (or, t→ ∞), we mean the existence of the limit (in the weak^* sense) μ:=limt→∞ μt. We present a large family of non-trivial examples of f where the selection of measure exists. These f belong to a sub-class of potentials introduced by P. Walters. In this case, explicit expressions for the selected V can be obtained for a certain large family of potentials.