Statistical properties of complex systems with subexponetnatial instability and phase transition

这个项目的目的之一是阐明动力学的次指数不稳定性如何影响复杂系统的统计特性。特别是，我们将我们的注意力集中在显示相变的非双曲现象上。为此，考虑了具有与给定势函数相关的广义不同周期轨道的分段可逆系统。对于这类系统，文献[1]指出，这些轨道的存在导致平衡态(相变)的非唯一性和平衡测度的非吉布斯性质。更具体地说，我们在文[1]中建立了平衡态在Bowen意义下的非吉布斯行为，压力函数(相变)的不可微性，双曲区域上停止时间分布的类似幂的尾巴，以及与点态维度相关的水平集的Hausdorff维度。特别地，压力函数的不可微性与多重分形问题有关。我们在文[2]中还证明了不变遍历弱Gibbs测度相对于弱Gibbs共形测度绝对连续的自然扩张具有某种形式的u-Gibbs性质。特别地，如果动态势允许广义不同的周期点，那么自然扩张在统计力学中表现出非吉布斯性质。这个项目的另一个目的是在Dobrushin意义下将间歇映射的非Gibbsian弱Gibbs度量与统计力学中的非Gibbsian弱Gibssian态联系起来。这一目的是在文献[31]中实现的，我们证明了一个高维间歇映射，其中Sinai-Bowen-Ruelle测度是弱Gibbs平衡态和Dobrushin意义下的弱Gibbsian态，Dobrushin允许在其条件概率中存在本质的间断。

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.