Cardinalities Of Scrambled Sets And Positive Scrambled Sets

In the part of introduction of the paper, the dynamics and chaos arc intro-ducted.In the first chapter of this paper, some basic definitions are introducted. And some results about scrambled sets and the cardinality of scrambled sets are summarized. At last, some question are raised.The second chapter of this paper, we study scrambled sets and positive scrambled sets of maps, and discuss the existance of maps possessing scrambled sets and positive scrambled sets with given cardinalities. The main results are as follows,Theorem 2.2.5. There exists a docile right-shift homeomorphisms f_k from I² × I² to itself for any k ∈ N, such that f_k have a 1—scrambled set containing k + 1 points but has no scrambled set containing more than k + 1 points.Theorem 2.2.6. There exists a docile right-translation homeomorphism f_a from I² × I² to itself such that f_a have a 1—scrambled set with cardinality a but has no uncountable scrambled set.Theorem 2.2.7. There exists a docile right-translation homeomorphism f_c0 from I² × I² to itself, such that the whole semi-open interval {1/2} × [0,1) is a scrambled of f_c0, but f_c0 has neither scrambled set with greater cardinality than c nor countably infinite positive scrambled set.Theorem 2.2.8. There exists a docile right-translation homeomorphism f(c1) from I² × I² to itself such that the whole semi-open interval {1/2} × [0,1) is a 1—scrambled of f_c1.Theorem 2.3.1. There exists locally path connected contractible metric space X with diameter 1 and a homeomorphism f on X for any given greater cardinality e than c, such that f has a 1— scrambled set with cardinality e, but/ has no scrambled set with greater cardinality than e.Theorem 2.3.2. Let cardinalities h > g > a. If there exist countably many less cardinalities than g {e; : i G Z+}, which needn't be different each other, such that the product of these cardinalities YliLi ei ^ h, then there exists a metric space X and a homeomorphism from X to itself such that / has a scrambled set with cardinality h, but / has neither positive scrambled set with cardinality g nor scrambled set with greater cardinality than h.Theorem 2.3.4. Let cardinalities h > g > a. If YiiLi ?i < h for arbitrarily countable less cardinalities {ej : i € Z+} than g, which needn't be different each other, then any continuous map / onto itself on any metric space X has a positive scrambled set with cardinality g if / has a scrambled set with cardinality h.Theorem 2.3.7. Assume that KQ is a a—power invariant cardinality, / is a continuous map from X to itself and nEN. / has a positive scrambled set with cardinality HQ+n if / has a scrambled set with cardinality HQ+ra.Theorem 2.3.9. If a continuous map / from a metric space X to itself has a scrambled set with greater cardinality than c, then / has a positive scrambled set with greater cardinality than c.