On statistical properties for nonlinear nonhyperbolic systems

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(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.