  Consider the semi-flow given by the continuous time shift
$\Theta_t:\mathcal{D} \to \mathcal{D} $, $t \geq 0$, acting on the
  $\mathcal{D} $ of \textit{c\`{a}dl\`{a}g} paths $w: [0,\infty) \to S^1$,
where $S^1$ is the unitary circle. We equip the space $\mathcal{D} $ with the
Skorokhod metric, and we show that the semi-flow is expanding. We also
introduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ acts
linearly on continuous functions $f:S^1\to\mathbb{R}$. This stochastic
semigroup and an initial vector of probability $\pi$ define an associated
stationary shift-invariant probability $\mathbb{P}$ on the Polish space
$\mathcal{D} $. Given such $\mathbb{P}$ and an H\"older potential $V:S^1 \to
\mathbb{R}$, we define a continuous time Ruelle operator, which is described by
a family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuous
functions $\varphi: S^1 \to \mathbb{R}$. More precisely, given any H\"older $V$
and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by
  $\varphi \to \psi(y) = \mathbb{L}^t_V(\varphi)(y)= \int_{w(t)=y} e^{ \int_0^t
V(w(s)) ds} \varphi (w(0)) d \mathbb{P}(w).$
  For some specific parameters we show the existence of an eigenvalue
$\lambda_V$ and an associated H\"older eigenfunction $\varphi_V>0$.After a
coboundary procedure we obtain another stochastic semigroup, with infinitesimal
generator $L_V$, and this will define a new probability $\mathbb{P}_V$ on
$\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for the
potential $V$. In this case, we define entropy for some shift-invariant
probabilities on $\mathcal{D}$, and we consider a variational problem of
pressure. Finally, we define entropy production and present our main result: we
analyze its relation with time-reversal and symmetry of $L$. We also show that
the continuous-time shift $\Theta_t$, acting on the Skorohod space $D$, is
expanding.
