On statistical properties for nonlinear nonhyperbolic systems

(1997)的目的之一是建立非一致双曲性系统的大偏差结果。对于这个问题，我解决了如下问题：对于具有一个不同周期轨道的分段C^1-Bernoulli映射，在一定的条件下，更准确地说，在水平2大偏差原理有界积分的上界下，建立了按导数加权的前象的大偏差结果。为了解决这一问题，波利科特教授对札幌的访问是有意义的。更重要的是，我可以获得一种积极的感觉，在未来建立非双曲映射的相关界限。本项目的另一个目的是研究遍历和的渐近性，特别是收敛于正态分布。在这个问题上，M.Denker教授的访问为解决这个问题做出了很大贡献。事实上，通过他的建议，我可以对我的例子的重要性有信心，我可以建立中心极限定理的一个推广结果，它适用于新的非双曲现象。包含这些结果的论文[4]提交给美国数学学会会刊。(1998)在第二年的项目中，我研究了具有不同周期点的映射的相关衰减率。我可以提出一种方法，通过推广利瓦尼的Perron-Frobenius算子的随机扰动来估计速率，这可以追溯到Bunimovich-Sinai-Chernov的马尔可夫近似方法。这一结果包含在与M.Pollicott联合发表的论文[5]中。

(1997)One of purposes of this project is to establish Large Deviation results for systems without uniformly hyperbolicity. About this problem, I solved it as follows : For piecewise C^1-Bernoulli maps with one indifferent periodicorbits, Large Deviation results for preimages weighted by the derivativeswere established under certain conditions, more precisely, upper boundsin the level 2 Large Deviation Principle <bounded integral>. To solve the problem, Prof.M.Pollicott's visit to Sapporo was meaningful. Further more I could obtain a positive feeling to establish bounds on correlations for nonhyperbolic maps in future. Another purpose of this project is to study asymptotic behavior of ergodic sums, in particuler, convergence to thenormal distributions.About this problem, Prof.M.Denker's visit gave a lot of contributions in solving it. In fact, by his advices I could have a confidenceof importance of my examples and I could establish an extended resultson the central limit theorem which are ajplicable to new nonhyperbolicphenomena. The paper [4] containing the results was submitted to Transaction of the American mathematical Society.(1998)In the second year project, I studied the rates of decay ofcorrelationsfor maps with indifferent perodic points. I could present an approach to estimating the rates by generalizing Liverani's random parturbations of Perron-Frobenius operators which goes back to Bunimovich-Sinai-Chernov's Markov approximation method. The result is contained in [5] which is a joint paper with M.Pollicott.