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

Abstract

Consider the semi-flow given by the continuous time shift $\hat{I}{˜}_t:\mathcal{D}\to \mathcal{D}$, $t\ge 0$, acting on the $\mathcal{D}$ of \textit{cà dlà 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 $e^{tL}$, $t\ge 0,$ where $L$ acts linearly on continuous functions $f:S^1\to \mathbb{R}$. This stochastic semigroup and an initial vector of probability $\ddot{I}€$ define an associated stationary shift-invariant probability $\mathbb{P}$ on the Polish space $\mathcal{D}$. Given such $\mathbb{P}$ and an Hölder potential $V:S^1\to \mathbb{R}$, we define a continuous time Ruelle operator, which is described by a family of linear operators ${\mathbb{L}}_V^t$, $t\ge 0,$ acting on continuous functions $\ddot{I}†:S^1\to \mathbb{R}$. More precisely, given any Hölder $V$ and $t\ge 0$, the operator ${\mathbb{L}}_V^t$, is defined by $\ddot{I}†\to \ddot{I}ˆ(y)={\mathbb{L}}_V^t(\ddot{I}†)(y)={\int }_{w(t)=y}{e}^{{\int }_0^{t}V(w(s))ds}\ddot{I}†(w(0))d\mathbb{P}(w).$ For some specific parameters we show the existence of an eigenvalue $\hat{I}{»}_V$ and an associated Hölder eigenfunction $\ddot{I}{†}_V>0$.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 $\hat{I}{˜}_t$, acting on the Skorohod space $D$, is expanding.

We made some changes to the material from the last submission. See new Appendix 3