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

Consider the semi-flow given by the continuous time shift Θ_t:D→D, t≥0, acting on the D of càdlàg paths (right continuous with left limits) w:[0,∞)→S¹, where S¹ is the unitary circle (one can also take [0, 1] instead of S¹). We equip the space D with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group e^tL, t≥0, where L (the infinitesimal generator) acts linearly on continuous functions f:S¹→R. This stochastic semigroup and an initial vector of probability π define an associated stationary shift-invariant probability P on the Polish space D. This probability P will play the role of an a priori probability. Given such P and an Hölder potential V:S¹→R, we define a continuous time Ruelle operator, which is described by a family of linear operators L_V^t, t≥0, acting on continuous functions φ:S¹→R. More precisely, given any Hölder V and t≥0, the operator L_V^t, is defined by (Formula presented.) For some specific parameters we show the existence of an eigenvalue λ_V and an associated Hölder eigenfunction φ_V>0 for the semigroup L_V^t, t≥0. After a coboundary procedure we obtain another stochastic semigroup, with infinitesimal generator L_V, and this will define a new probability P_V on D, which we call the Gibbs (or, equilibrium) probability for the potential V. In this case, we define entropy for some continuous time shift-invariant probabilities on 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 Θ_t, acting on the Skorokhod space D, is expanding. We wonder if the point of view described here provides a sketch (as an alternative to the Anosov one) for the chaotic hypothesis for a particle system held in a nonequilibrium stationary state, as delineated by Ruelle, Gallavotti, and Cohen.