Measure theoretical approach for chaotic dynamical systems
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概要
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この論文は国立情報学研究所の電子図書館事業により電子化されました。We study chaotic dynamical systems from measure theoretical view pohrt. Let f be a observable defined on a phase spuce X. We are interested in tirne evolutions of it f, fT, fT^2, …, fT^n…, i.e., their asymptotic distribution behaviors and limiting distributions of their partial sums. In ergodic theory, there are several characterizations of dynamical systems with high degrees bf randomness. Typical examples are hyperbolic systems which can be reduced to Markov process through a nice coarse-graining. On the other hand, in probability, weak independent process are studied by many people. However most of those are established only for Markov processes and strongly independent processes. In this talk, we introduce a technique for connecting random process come from dynamical systems and random process in probability theory. It allows us to ch8racterize sensitive dependence of dynamical systems on initial distributions. Further we show several chaotic dynamical systems in the above sense which are not necessarily hyperbolic. One of such Mathematical models, a class of mappings providing nice measure theoretical structure is the following : Let X be a bounded domain of B^d and T a transformation of X. Assume that there exists a generating countable partition Q={X_a}a∈I of X s.t. T|X_a : X_a→TX_a is a C^1_diffeomorphism. We define a cylinder set of rank n by X_<a1>∩T^<-1>X_a2∩…∩T^<-(n-1)>X_<an-1> if its interior is not empty and we denote by X_<a1…an>. Let U={T^Nx_<a1…n> : VX_<a1…an>, Vn>0}. If U is a finite set, we call the quadruple (T, X, Q={X_a}, U) a piecewise invertible system with finite range structure (FRS). Piecewise exapanding Markov maps are particular cases of the above system. Typical examples are number theoretical transformations.
- 物性研究刊行会の論文
- 1996-07-20
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