Non-Gibbsianness and phase transition in complex systems

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The main purpose of this project is to clarify typical reasons for phase transition and non-Gibbsianness of equilibrium measures in the context of complex systems. For this purpose, we first established large deviation properties for countable to one Markov systems associated with weak Gibbs measures for non-H"older potentials. We clarified a class of functions in which we can describe free energy function associated to weak Gibbs measures in terms of topological pressure. We established the level-2 upper large deviaition inequality and clarified sufficient conditions for the upper bounds being strictly negative. Furthermore, we studied multifractal large deviation laws for non-hyperbolic systems exhibiting 'intermittency'. In particular, the law is established for countable to one piecewise conformal Markov systems, which are derived systems constructed over hyperbolic regions. We formulated different stages of 'indifferency ' associated with potentials of weak bounded variation, and … More relate new characterization of phase transitions to indifferent periodic points at various stages. We also succeeded to associate non-differentiability of the Hausdorff dimension of level sets with phase transitions for intermittent systems. These results were established in [1]. The second purpose of this project is to describe dissipative phenomena via invertible extensions of non-invertible non-hyperbolic systems. We restrict our attention to countable to one sofic systems and established an explicit topological description of spaces of their invertible extensions under the existence of their dual systems. We also give a topological notion of ' reversibility ' of the invertible extensions. Then we discuss when Rohlin's invertible extension of ergodic absolutely continuous invariant measures are absolutely continuous with respect to a natural physical measure on the spaces that we constructed. We could observe that different nature of observability between original non- invertible systems and their dual systems causes dissipative phenomena in their invertible extensions. Those results are obtained in [4]. The third purpose of this project is to introduce thermodynamic methods to study conformal measures for families of partially defined maps on compact metric spaces which is called a ' dynamical family '. In particular, we could develop some aspects of thermodynamic formalism for countable to one sofic systems. Those results will appear in a joint paper with M.Denker (Gottingen Univ.). Less