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

Abstract

Denote by $\bm{\mu}$ the maximal entropy measure for the shift \(\sigma\) acting on $\Omega = \{0, 1\}^\mathbb{N}$, by $\ruelle$ the associated Ruelle operator and by $\koopman = \ruelle^{\dagger}$ the Koopman operator, both acting on $\lp{2}(\bm{\mu})$. 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 \(\lp{p}\)-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 \(\lp{p}\) context, \(p \geq 1\). Given the states $\eta, \xi$: $d_{p}(\eta, \xi) \defn \sup \{ \,|\eta(a) - \xi(a) | where a \in \mathcal{A} and \norm{\left[\mathcal{D}_p,\pi(a)\right]} \leq 1\}$. The operator $M_f$ acts on $L^p (\mu).$ We explore the relationship of $\mathcal{D}_p$ with dynamics, in particular with $f \circ \sigma - f$, the discrete-time derivative of a continuous $f:\Omega \to \mathbb{R}$. Take $p,p^{\prime}>0$ satisfying $\frac{1}{p} + \frac{1}{ p^{\prime}}=1$. We show for any continuous function $f$: $\norm{\left[ \dirac_p, \pi(\mult_f) \right]} = | \sqrt[\lambda]{\ruelle \abs{f \circ \sigma - f}^{\lambda}} |_{\infty}$, where $\lambda = \max\{p, p^\prime\}$. Furthermore, we show $\norm{\left[ \mathcal{D}_p, \pi(\koopman^{n} \mathcal{L}^{n})]\right]}=1$ for all \(n \geq 1\). We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.