The Dirac operator for the Ruelle-Koopman pair on L^p-spaces: an interplay between Connes distance and symbolic dynamics

Denote by μ the maximal entropy measure for the shift \(σ\) acting on Ω = \0, 1, by the associated Ruelle operator and by = ^ the Koopman operator, both acting on 2(μ). Using a diagonal representation π, the Ruelle-Koopman pair can be used for defining a dynamical Dirac operator D, as in {BL}. D plays the role of a derivative. In {lpspec}, the notion of a spectral triple was generalized to \({p}\)-operator algebras; in consonance, here, we generalize results for D to results for a Dirac operator Dₚ , and the associated Connes distance dₚ, to this new \({p}\) context, \(p ≥ 1\). Given the states η, ξ: dₚ(η, ξ) \ \,|η(a) - ξ(a) | where a ∈ A and [Dₚ,π(a)] ≤ 1\. The operator Mf acts on Lᵖ (μ). We explore the relationship of Dₚ with dynamics, in particular with f ∘ σ - f, the discrete-time derivative of a continuous f:Ω → R. Take p,p^>0 satisfying 1/p + 1 p^=1. We show for any continuous function f: [ ₚ, π(f) ] = | √[λ] f ∘ σ - fλ |∞, where λ = max, p^\. Furthermore, we show [ Dₚ, π(ⁿ Lⁿ)]]=1 for all \(n ≥ 1\). We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.