Non-Gibbsianness and phase transition in complex systems

这个项目的主要目的是澄清在复杂系统的背景下，平衡措施的相变和非吉布斯的典型原因。为此，我们首先建立了非H“older势的可数到一马氏系统的弱Gibbs测度的大偏差性质。我们阐明了一类可以用拓扑压来描述弱Gibbs测度的自由能函数的函数。建立了二阶上大偏差不等式，并阐明了上界严格为负的充分条件.此外，我们还研究了非双曲型方程组的多重分形大偏差律。特别地，建立了可数到一分段共形马尔可夫系统，这是双曲区域上构造的导出系统的法律。我们制定了与弱有界变差势相关的“无差异”的不同阶段， ...更多信息 将相变的新特征与各个阶段的无关周期点联系起来。我们还成功地将水平集的Hausdorff维数的不可微性与间歇系统的相变联系起来。这些结果在[1]中得到了证实。这个项目的第二个目的是通过不可逆非双曲系统的可逆扩展来描述耗散现象。我们把注意力限制在可数到一个sofic系统，并建立了一个明确的拓扑描述的空间，其可逆扩张的存在下，其对偶系统。我们还给出了可逆扩张的“可逆性”的拓扑概念。然后讨论了遍历绝对连续不变测度的Rohlin可逆扩张在所构造的空间上关于自然物理测度绝对连续的条件。我们可以观察到，原不可逆系统与其对偶系统的可观测性不同，导致了其可逆扩张中的耗散现象。这些结果在[4]中得到。这个项目的第三个目的是引入热力学方法来研究紧致度量空间上的部分定义映射族的共形测度，这类映射族被称为“动力学族”。特别地，我们可以发展可数到一个sofic系统的热力学形式主义的某些方面。这些结果将出现在与M.Denker（哥廷根大学）的联合论文中。少

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