论文标题
Hartman-Grobman定理中同构的较高规律性和对其清晰度的猜想
Higher regularity of homeomorphisms in the Hartman-Grobman theorem and a conjecture on its sharpness
论文作者
论文摘要
Hartman-Grobman定理指出，在适当的假设下，同构h将非线性系统的解决方案发送到其线性化的解决方案。许多数学家已做出贡献，以证明同态同态的连续性。但是，是否有可能改善Hölder连续性与Lipschitzian连续性？ 本文给出了积极的答案。我们提出了第一个结果，即同构是Lipschitzian，而不是$ C^1 $，而其逆性仅是Hölder的连续，而不是Lipschitzian。有趣的是，同构的规律性不同于其反向。此外，提出了一些说明性示例，以显示我们结果的有效性。此外，在我们的例子的驱动下，我们还提出了一个猜想，说同构的规律性很敏锐，无法再改善。
Hartman-Grobman theorem states that there is a homeomorphism H sending the solutions of the nonlinear system onto those of its linearization under suitable assumptions. Many mathematicians have made contributions to prove Hölder continuity of the homeomorphisms. However, is it possible to improve the Hölder continuity to Lipschitzian continuity? This paper gives a positive answer. We formulate the first result that the homeomorphism is Lipschitzian, but not $C^1$, while its inverse is merely Hölder continuous, but not Lipschitzian. It is interesting that the regularity of the homeomorphism is different from its inverse. Moreover, some illustrative examples are presented to show the effectiveness of our results. Further, motivated by our example, we also propose a conjecture, saying, the regularity of the homeomorphisms is sharp and it could not be improved any more.
