We study the continuity of topological entropy of general diffeomorphisms on line. First, we prove that the entropy map is continuous with respect to the strong $C^{0}$-topology on the union of uniformly topologically hyperbolic diffeomorphisms contained in $\text{Diff}_{0}^{r}(\mathbb{R})$ (whose first derivative is uniformly away from zero), which is a $C^{0}$-open and $C^{r}$-dense subset of $\text{Diff}_{0}^{r}(\mathbb{R})$, $r=1,2,\ldots ,\infty$, and $\unicode[STIX]{x1D714}$ (real analytic). Second, we give some examples where entropy map is not continuous. Finally, we prove some results on the continuity of entropy of general diffeomorphisms on the (real) line.

中文翻译：

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线上一般微分同胚的熵

我们在线研究一般微分同胚的拓扑熵的连续性。首先，我们证明熵图对于强$C^{0}$-关于统一拓扑双曲微分同胚并集的拓扑$\text{Diff}_{0}^{r}(\mathbb{R})$（其一阶导数一致地远离零），这是$C^{0}$-打开和$C^{r}$-密集子集$\text{Diff}_{0}^{r}(\mathbb{R})$,$r=1,2,\ldots ,\infty$， 和$\unicode[STIX]{x1D714}$（真正的分析）。其次，我们给出一些熵图不连续的例子。最后，我们证明了一般微分同胚在（实）线上的熵连续性的一些结果。