The Dirac operator for the Ruelle-Koopman pair on L^p-spaces: an interplay between Connes distance and symbolic dynamics
William Braucks, Artur O. Lopes
TL;DR
The article develops a dynamical Dirac operator $D_p$ for the Ruelle–Koopman pair on $L^{p}$ spaces over a symbolic space, enabling a Connes distance framework in a dynamical setting. It establishes explicit commutator norms in terms of the discrete-time derivative $f\circ\sigma - f$ and the transfer operators, proving that $\lVert[D_p, \pi(M_f)]\rVert$ can be characterized by a maximum over $p$-norms of weighted Koopman–Ruelle actions. The work analyzes the Connes distance for both pure and general states, deriving finiteness criteria tied to homoclinic equivalence and linking the distance to Wasserstein-type transport costs, with $d_p$ interpolating between Wasserstein distances and the Connes metric as $p$ varies. It also demonstrates that $\lVert[D_p, \pi(K^nL^n)]\rVert = 1$ for all $n$ and provides concrete bounds for distances between state pairs, thereby bridging noncommutative geometry, symbolic dynamics, and noncommutative optimal transport.
Abstract
Denote by $\bmμ$ the maximal entropy measure for the shift \(σ\) acting on $Ω= \{0, 1\}^\mathbb{N}$, by $\ruelle$ the associated Ruelle operator and by $\koopman = \ruelle^{\dagger}$ the Koopman operator, both acting on $\lp{2}(\bmμ)$. Using a diagonal representation $π$, 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 $η, ξ$: $d_{p}(η, ξ) \defn \sup \{ \,|η(a) - ξ(a) | where a \in \mathcal{A} and \norm{\left[\mathcal{D}_p,π(a)\right]} \leq 1\}$. The operator $M_f$ acts on $L^p (μ).$ We explore the relationship of $\mathcal{D}_p$ with dynamics, in particular with $f \circ σ- f$, the discrete-time derivative of a continuous $f:Ω\to \mathbb{R}$. Take $p,p^{\prime}>0$ satisfying $\frac{1}{p} + \frac{1}{ p^{\prime}}=1$. We show for any continuous function $f$: $\norm{\left[ \dirac_p, π(\mult_f) \right]} = | \sqrt[λ]{\ruelle \abs{f \circ σ- f}^λ} |_{\infty}$, where $λ= \max\{p, p^\prime\}$. Furthermore, we show $\norm{\left[ \mathcal{D}_p, π(\koopman^{n} \mathcal{L}^{n})]\right]}=1$ for all \(n \geq 1\). We also prove a formula analogous to the Kantorovich duality formula for minimizing the cost of tensor products.