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

Denote by $\mathit{\mu }$ the maximal entropy measure for the shift $\sigma$ acting on $\mathrm{\Omega }=\{0,1{\}}^{\mathbb{N}}$, by $\text{ruelle}$ the associated Ruelle operator and by $\text{koopman}={\text{ruelle}}^{†}$ the Koopman operator, both acting on $\text{lp}2(\mathit{\mu })$. Using a diagonal representation $\pi$, the Ruelle-Koopman pair can be used for defining a dynamical Dirac operator $\mathcal{D},$ as in \cite{BL}. $\mathcal{D}$ plays the role of a derivative. In \cite{lpspec}, the notion of a spectral triple was generalized to $\text{lp}p$-operator algebras; in consonance, here, we generalize results for $\mathcal{D}$ to results for a Dirac operator ${\mathcal{D}}_p$ , and the associated Connes distance $d_p$, to this new $\text{lp}p$ context, $p\ge 1$. Given the states $\eta ,\xi$: $d_p(\eta ,\xi )\text{defn}\sup \{\mathrm{\mid }\eta (a)-\xi (a)\mathrm{\mid }wherea\in \mathcal{A}and\text{norm}[{\mathcal{D}}_p,\pi (a)]\le 1\}$. The operator $M_f$ acts on $L^p(\mu )\mathrm{.}$ We explore the relationship of ${\mathcal{D}}_p$ with dynamics, in particular with $f\circ \sigma -f$, the discrete-time derivative of a continuous $f:\mathrm{\Omega }\to \mathbb{R}$. Take $p,p^{\mathrm{\prime }}>0$ satisfying $\frac{1}{p}+\frac{1}{p^{\mathrm{\prime }}}=1$. We show for any continuous function $f$: $\text{norm}[{\text{dirac}}_p,\pi ({\text{mult}}_f)]=\mathrm{\mid }\sqrt[\lambda ]{\text{ruelle}\text{abs}{f\circ \sigma -f}^{\lambda }}{\mathrm{\mid }}_{\mathrm{\infty }}$, where $\lambda =\max \{p,p^{\mathrm{\prime }}\}$. Furthermore, we show $\text{norm}[{\mathcal{D}}_p,\pi ({\text{koopman}}^n{\mathcal{L}}^n)]]=1$ for all $n\ge 1$. We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.