  Let $p\in [1,\infty)$. We define an $L^p$-operator algebra crossed product by
a transfer operator for the topological Bernoulli shift $\varphi$ on
$X=\{1,...,n\}^{\mathbb{N}}$, and we prove it is isometrically isomorphic to
the $L^p$-analog $\mathcal{O}_n^p$ of the Cuntz algebra introduced by Phillips.
As an application, we prove that the spectrum of the associated `abstract
weighted shift operators' $aT$, $a\in C(X)$, is a disk with a radius given by
the formula $$ r(aT)=\max_{\mu\in\text{Erg}(X,\varphi)} \exp\left( \int_X
\ln(|a|\varrho^{\frac{1}{p}})d\mu + \frac{h_\varphi(\mu)}{p} \right) $$ where
$\varrho$ is a potential associated to the transfer operator, and
$h_\varphi(\mu)$ is Kolmogorov-Sinai entropy. This generalizes classical
results for $p=2$.
