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.

Este artículo explora los viajes en el tiempo y sus implicaciones.

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