Statistical properties of nonstationary weak Gibbs states and analysis of dissipative phenomena for those invertible extensions

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For nonhyperbolic, noninvertible piecewise invertible systems , we constructed natural extensions of both (bi)nonsingular measures and absolutely continuous invariant weak Gibbs measures under the finite range structure condition. Furthermore, we obtained a criterion for deciding the processes arising from those invertible extensions exhibit dissipation and new phenomena arising from intermittent sofic systems preserving weak Gibbs measures, which are not observed near equilibrium, that is dissipation of the phase volume with zero asymptotic averaged entropy production and bottomless source of the Gibbs entropy with zero asymptotic averaged entropy production. For families of partially defined maps on compact metric spaces, we clarified when self-homeomorphic sets can be determined. In particular, under sofic condition we gave a sufficient condition for potential functions admitting conformal measures.