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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 theSkorokhod metric, and we show that the semi-flow is expanding. We alsointroduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ actslinearly on continuous functions $f:S^1\to\mathbb{R}$. This stochasticsemigroup and an initial vector of probability $\pi$ define an associatedstationary 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 bya family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuousfunctions $\varphi: S^1 \to \mathbb{R}$. More precisely, given any H\"older $V$and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by  $\varphi \to \psi(y) = \mathbb{L}^t_V(\varphi)(y)= \int_{w(t)=y} e^{ \int_0^tV(w(s)) ds} \varphi (w(0)) d \mathbb{P}(w).$  For some specific parameters we show the existence of an eigenvalue$\lambda_V$ and an associated H\"older eigenfunction $\varphi_V>0$.After acoboundary procedure we obtain another stochastic semigroup, with infinitesimalgenerator $L_V$, and this will define a new probability $\mathbb{P}_V$ on$\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for thepotential $V$. In this case, we define entropy for some shift-invariantprobabilities on $\mathcal{D}$, and we consider a variational problem ofpressure. Finally, we define entropy production and present our main result: weanalyze its relation with time-reversal and symmetry of $L$. We also show thatthe continuous-time shift $\Theta_t$, acting on the Skorohod space $D$, isexpanding.

Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness