Statistical properties of equilibrium states for complex systems

（1999）1999年该项目的目的之一是阐明可数到一分段可逆马尔可夫系统存在共形测度的充分条件。关于这个问题，M.Denker教授在札幌期间提出了宝贵的建议，使我能够建立一种新的构造共形测度的方法，这种方法是基于存在一个导出的映射T^*（Schweiger跳跃变换），它是一致膨胀的，并保证了与φ相关联的势φ^* 的弱Holder型性质。结果包含在（3）中，这是与M. Denker的联合论文。该项目的另一个目的是建立边界的衰减相关函数的不可逆映射与中立的周期点。应用基于Perron-Frobenius算子随机扰动的Liverani方法，并通过对文献[1]中建立的不变密度在不同周期点附近的发散阶的估计，得到了多项式的界。结果包含在与M. Pollicott的联合文章（4）中。（2000）在第二年的项目中，我研究了具有中性周期点的不可逆映射的动态zeta函数的亚纯性质。我可以通过观察φ的拓扑压力和与φ^* 相关的关于跳跃变换T^* 的拓扑压力之间的良好关系，来阐明zeta函数的亚纯dmeain。结果包含在（5）中，这是与M. Pollicott的联合论文。此外，我可以通过澄清迭代Perron-Frobenius算子在不包括无关周期点的紧集上的一致收敛速度来改进（4）中关于相关性衰减速率的结果。结果包含在（6）中。

(1999)One of purposes of the project in 1999 is to clarify sufficient conditions for the existence of conformal measures for countable to one piecewise invertible Markov systems. About this problem, Prof.M.Denker gave valuable advices during his stay in Sapporo so that I could establish a new method for the construction of conformal measures which is based on the existence of a derived map T^* (Schweiger's jump transformation) which is uniformly expanding and guarantees a weak Holder-type property of the potential φ^* associated to φ. The result is contained in (3) which is a joint paper with M.Denker. Another purpose of the project is to establish bounds on decay of correlation functions for noninvertible maps with indifferent periodic points. I could obtained polynomial bounds by applying Liverani's method based on random parturbations of Perron-Frobenius operators and by estimating order of divergence of invariant densities near indifferent periodic points which was established in (1). The result is containaed in (4) which is a joint paper with M.Pollicott.(2000)In the second year project, I studied meromorphic properties of dynamical zeta functions for noninvertible maps with indifferent periodic points. I could clarify the meromorpic dmeain of the zeta functions by observing a good relation between the topological pressure for φ and the topological pressure associated to φ^* with respect to the jump transformation T^*. The result is contained in (5) which is a joint paper with M.Pollicott. Furthermore, I could improve the results on the rates of decay of correlations in (4) by clarifying the speed of uniform convergence of iterated Perron-Frobenius operators on compact sets excluding indifferent periodic points. The result is contained in (6).