The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological space X satisfying the second axiom of countability and for an outer measure m on X satisfying the cond...The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological space X satisfying the second axiom of countability and for an outer measure m on X satisfying the conditions: (i) every non-empty open set of X is m-measurable with positive m-measure; (ii) the restriction of m on Borel σ-algebra B( X) of X is a probability measure, and (iii) for every Y X there exists a Borel set B B(X) such that B Y and m (B)= m (Y), if f : X→X is a strong-mixing measure-preserving transformation of the probability space (X,B(X), m), and if {mi} is a strictly increasing sequence of positive integers, then there exists a subset C X with m (C) = 1, finitely chaotic with respect to the sequence {mi}, i e. for any finite subset A of C and for any map F:A→X there is a subsequence {ri} such that limt→∞fri(a) = F(a) for any a∈A . There are some applications to maps of one展开更多
We discuss a kind of measure-preserving mappings T related to the Couette-Taylor system where A, B, C, D, E, F are parameters. This is a rather particular 3-dimensional measure-preserving mapping with existence of inv...We discuss a kind of measure-preserving mappings T related to the Couette-Taylor system where A, B, C, D, E, F are parameters. This is a rather particular 3-dimensional measure-preserving mapping with existence of invariant curves(1-dimensional invariant manifolds) in the neighbourhood of a fixed point. The remits show that the ordered region will decrease when the perturbation parameters C, D, E, F increase and display the behaviour of mapping T at different distances from the fixed point.展开更多
In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant cur...In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.展开更多
In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant cur...In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.展开更多
For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assu...For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {γr}γ∈Γis a countable family of relations of the full measure on a probability space (X,B,μ), i.e. for everyγ∈Γthere is a positive integer sγsuch that Rγ(?)Xsγwithμsγ(Rγ) = 1. In the present paper we show that if (X, B,μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K(?)X withμ*(K) = 1 such that (x1,...,xsγ)∈γr for anyγ∈Γand for any sγdistinct elements x1,..., xsγof K, whereμ* is the outer measure induced by the measureμ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.展开更多
基金Project supported by the National Natural Science Foundation of China.
文摘The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological space X satisfying the second axiom of countability and for an outer measure m on X satisfying the conditions: (i) every non-empty open set of X is m-measurable with positive m-measure; (ii) the restriction of m on Borel σ-algebra B( X) of X is a probability measure, and (iii) for every Y X there exists a Borel set B B(X) such that B Y and m (B)= m (Y), if f : X→X is a strong-mixing measure-preserving transformation of the probability space (X,B(X), m), and if {mi} is a strictly increasing sequence of positive integers, then there exists a subset C X with m (C) = 1, finitely chaotic with respect to the sequence {mi}, i e. for any finite subset A of C and for any map F:A→X there is a subsequence {ri} such that limt→∞fri(a) = F(a) for any a∈A . There are some applications to maps of one
文摘We discuss a kind of measure-preserving mappings T related to the Couette-Taylor system where A, B, C, D, E, F are parameters. This is a rather particular 3-dimensional measure-preserving mapping with existence of invariant curves(1-dimensional invariant manifolds) in the neighbourhood of a fixed point. The remits show that the ordered region will decrease when the perturbation parameters C, D, E, F increase and display the behaviour of mapping T at different distances from the fixed point.
文摘In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.
文摘In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.
基金This work was supported by the National Science Fbundation of China (Grant No. 10471049)
文摘For a probability space (X, B,μ) a subfamily F of theσ-algebra B is said to be a regular base if every B∈B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {γr}γ∈Γis a countable family of relations of the full measure on a probability space (X,B,μ), i.e. for everyγ∈Γthere is a positive integer sγsuch that Rγ(?)Xsγwithμsγ(Rγ) = 1. In the present paper we show that if (X, B,μ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K(?)X withμ*(K) = 1 such that (x1,...,xsγ)∈γr for anyγ∈Γand for any sγdistinct elements x1,..., xsγof K, whereμ* is the outer measure induced by the measureμ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.
