2014/04/30 by Jairo K. Mengue, Mengue, Jairo K., Elismar R. Oliveira +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Engineering · Mathematics · Physics and Astronomy · #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #math-ph #math.DS #math.MP #math.OC
paper · pdf · doi:10.48550/arxiv.1404.7801
20 pages
openalex publication_date 2014/04/30 · arxiv created 2015/07/09 · arxiv updated 2015/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For X, Y, Z and W compact metric spaces, consider two uniformly contractive IFS \τx: Z→ Z, x∈ x\ and \τy:W→ W, y∈ Y\. For a fixed α∈ P(X) with supp(α)=X we define the entropy of a holonomic measure π∈ P(X× Z) relative to α, the pressure of a continuous cost function c(x,z) and show that for c Lipschitz this pressure coincides with the spectral radius of the associated transfer operator. The same approach can be applied to the pair Y,W. For fixed probabilities α∈ P(X) and β∈ P(Y) with supp(α)=X, supp(β)=Y we denote by Hα(π), π∈ Π(⋅,⋅,τ), the entropy of the (X,Z)-marginal of π relative to α and denote by Hβ(π), the entropy of the (Y,W)-marginal of π relative to β. The marginal pressure of a continuous cost function c ∈ C(X× Y × Z × W) relative to (α,β) will be defined by Pm(c) = supπ∈Π(⋅,⋅,τ) ∫ c dπ+ Hα(π) +Hβ(π) and we will show the following duality result: infPm(c -φ(x) -ψ(y))=0 ∫ φ(x) dμ+∫ ψ(y) dν= supπ∈Π(μ,ν,τ) ∫ c dπ+ Hα(π) +Hβ(π). When Z and W have only one point and the entropy is unconsidered this equality can be rewritten as the Kantorovich Duality for compact spaces X,Y and continuous cost -c: infc -φ(x) -ψ(y)≤ 0 ∫ φ(x) dμ+∫ ψ(y) dν= supπ∈Π(μ,ν) ∫ c dπ.