【摘要】 <正> 可扩同胚(expansive homeomorphism)这概念最早见于[12],后来有一系列文章(例如[1,2,6])详细讨论了这种同胚的拓扑动力学性质。可扩同胚可视为微分动力体系中Anosov微分同胚的一种推广[11],因而有关可扩同胚的讨论对于了解Anosov微分同胚将是有帮助的[7],可扩同胚在结构稳定性理论、遍历理论以及符号动力系统中时有应用[6,P.313]。更多还原

【Abstract】 A Self-homeomorphism f of a metric space (X, d) is called expansive if there exists a real number e>o such that for any two Points x, y∈X, x≠y, there is an integer n such that d(fnx, fny)>e. W. H. Gottschalk [3, P. 348] raised the question whether an n-cell admits such a homeomorphism. It is known [1] that there is no expansive homeomorphism on 1-cell (open or closed). W. Reddy [103 and R. K. Williams [14] attempted to construct expansive homeomorphisms on a open 2-cell B2 respectively. However, in the appendix of this paper it is proved that neither homeomorphism they provided is expansive. Therefore, the problem whether an open 2-cell admits such a homeomorphism is still open as yetIn ? of this paper an expansive homeomorphism is constructed on an open 2-cell B2, and another is given on an open solid cylinder B2 × I in ?. Finally, through the finite productive property of expansive homeomorphism [13], we obtain our main result: an open n-dim ball Bn admits an expansive homeomorphism for n≥2The expansiveness of the homeomorphisms given here both on the open ball B2 and on the open solid cylinder B2 ×I is established by using such a property of the rationalnumber set {5/2,5/4,15/8} that there exists a number ∈ (0, 1/2) such that for any thr∈real numbers C1, C2, C3 which are not simultaneously zero and for any nonnegative integer N there is an integer n≥N such that the fractional part of the real number(5/2)nC1 +(5/4)nC2+(15/8)nC3 is in (ε , 1-ε ) (proposition 3.4 in this paper)Remark: After the submission of this paper, it was brought to the attention of the present author that Masaharu KOUNO (On expansive homeomorphisms on manifolds, J. of Math. Soc. Japan Vol. 33 No 3(1981), 533-538) obtained from a wider point of view the same results as ours in this paper. His method is geometrical and is different from ours shown above.更多还原