Statistical properties of weak Gibbs measures for complex systems with nonhyperbolic periodic orbits

这个项目的目的是建立数学模型，显示相变，吉布斯性质的失效和相关性的缓慢衰减，可以观察到许多复杂的系统表现出间歇性。为此，我们建立了非双曲情况下非holder势的热力学形式。更具体地说，对于可数到一个可传递马尔可夫系统，我们构造了弱有界变分势φ的弱Gibbs测度ν，并证明了平衡态μ在φ上的存在性，与弱Gibbs测度ν([1])等价。我们看到，广义的无差异周期轨道导致相变、非吉本性，并迫使相关性衰减缓慢（[1],[4]）。应用这些结果，我们建立了与弱有界变差（[3]）势相关的弱吉布斯测度的多重分形形式。此外，我们建立了遍历弱Gibbs测度μ([2])的可逆扩展的一个版本的局部积结构（弱局部积结构）。关于弱Gibbs平衡测度的相关性衰减，通过阐明不含无关周期点的紧集中迭代转移算子的一致收敛速度，得到了一类包含所有分段Lipschitz函数（[4]）的函数的多项式上界。

The purpose of this project is to construct mathematical models which show phase transition, failure of the Gibbs property and slow decay of correlations, that can be observed for many complex systems exhibiting Intermittency. For this purpose, we establish Thermodynamic formalism for non-Holder potentials in nonhyperbolic situation. More specifically, for countable to one transitive Markov systems we construct conformal measures ν which are weak Gibbs measures for potentials φ of weak bounded variation, and show the existence of equilibrium states μ for φ equivalent to the weak Gibbs measures ν ([1]). We see that generalized indifferent periodic orbits cause phase transition, non-Gibbsianness and force the decay of correlations to be slow ([1], [4]). Applying these results allow us to establish a multifractal formalism of weak Gibbs measures associated to potentials of weak bounded variation ([3]). Furthermore, we establish a version of the local product structure (weak local product structure) for the invertible extension of ergodic weak Gibbs measures μ ([2]). About the decay of correlations of weak Gibbs equilibrium measures, polynomial upper bounds can be obtained for a class of functions which contains all piecewise Lipschitz functions ([4]), by clarifying the speed of uniform convergence of iterated transfer operators on compact sets excluding indifferent periodic points.