Statistical properties of complex systems with subexponetnatial instability and phase transition

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One of the purpose of this project is to clarify how statistical properties of complex systems is influenced by subexponential instability of the dynamics. In particular, we direct our attention to non-hyperbolic phenomena exhibiting phase transitions. For this purpose, piecewise invertible systems with generalized indifferent periodic orbits associated to a given potential function are considered. For such systems, it was shown in [1] that the presence of such orbits causes non-uniqueness of equilibrium states (phase transitions) and non-Gibbsianness of equilibrium measures. More specifically, we established in [1] that non-Gibbsian behaviour of equilibrium states in the sense of Bowen, non-differentiability of the pressure function (phase transiton), powerlike tails of the distribution of the stopping times over hyperbolic regions, and the Hausdorff dimension of level sets associated to pointwise dimension. In particular, non-differentiability of pressure functions is related to multifractal problem. We also established in [2] that the natural extensions of invariant ergodic weak Gibbs measures absolutely continuous with respect to weak Gibbs conformal measures possess a version of u -Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points, then the natural extensions exhibit non-Gibbsian character in statistical mechanics. Another purpose of this project is to associate non-Gibbsian weak Gibbs measures for intermittent maps to non-Gibbsian weakly Gibbssian states in statistical mechanics in the sense of Dobrushin. This purpose was achiedved in [ 31 and we showed a higher dimensional intermittent map of which Sinai-Bowen-Ruelle measure is a weak Gibbs equilibrium state and a weakly Gibbsian state in the sense of Dobrushin admitting essential discontinuities in its conditional probabilities.