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 dimension展开更多
Let I be an interval (i. e. a nontrivial connected subset of the real line), and let f:Ⅰ→Ⅰ be continuous mapping.The set P(f) of periodic points and the nonwandering set Ω(f) are defined as usual.
For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-emp...For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-empty open set of the symbolic space ∑N is 1), satisfying the condition for any non-empty subset A of the set C, and for any continuous map F: A→∑N there exists a strictly increasing sequence {rn} of positive integers such that the sequence {σ (x)} converges to F(x) for any x∈A. On the other hand, it is shown that in ∑N every chaotic set for σ has 1-dimensional Hausdorff measure 0.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 dimension
文摘Let I be an interval (i. e. a nontrivial connected subset of the real line), and let f:Ⅰ→Ⅰ be continuous mapping.The set P(f) of periodic points and the nonwandering set Ω(f) are defined as usual.
基金Project supported by the National Natural Science Foundation of China.
文摘For the shift a of the symbolic space ∑N there exists a subset (called a chaotic set for σ) C of ∑N whose Hausdorff dimension is 1 everywhere (i.e. the Hausdorff dimension of the intersection of C and every non-empty open set of the symbolic space ∑N is 1), satisfying the condition for any non-empty subset A of the set C, and for any continuous map F: A→∑N there exists a strictly increasing sequence {rn} of positive integers such that the sequence {σ (x)} converges to F(x) for any x∈A. On the other hand, it is shown that in ∑N every chaotic set for σ has 1-dimensional Hausdorff measure 0.
