Denote by the maximal entropy measure for the shift \ (\) acting on = \0, 1\N, by the associated Ruelle operator and by = ^ the Koopman operator, both acting on 2 (). Using a diagonal representation, 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,: d (, ) \ \, | (a) - (a) | where a A and {Dₚ, (a) 1\}. The operator Mf acts on Lᵖ (). We explore the relationship of Dₚ with dynamics, in particular with f - f, the discrete-time derivative of a continuous f: R. Take p, p^>0 satisfying 1p + 1 p^{}=1. We show for any continuous function f: ₚ, (f) = | f - f^{} |_, where = \p, p^\. Furthermore, we show Dₚ, (^{n L^n) ]}=1 for all \ (n 1\). We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.
Braucks et al. (Mon,) studied this question.
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