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Entropy of diffeomorphisms of line

The author is supported by National Natural Science Foundation of China (grant no. 11501371,11671025), and Science Foundation of Shanghai Normal University (SK201502)
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  • For diffeomorphisms of line, we set up the identities between their length growth rate and their entropy. Then, we prove that there is $C^0$-open and $C^r$-dense subset $\mathcal{U}$ of $\text{Diff}^r (\mathbb{R})$ with bounded first derivative, $r=1,2,\cdots$, $+\infty$, such that the entropy map with respect to strong $C^0$-topology is continuous on $\mathcal{U}$; moreover, for any $f \in \mathcal{U}$, if it is uniformly expanding or $h(f)=0$, then the entropy map is locally constant at $f$.

    Also, we construct two examples:

    1. there exists open subset $\mathcal{U}$ of $\text{Diff}^{\infty} (\mathbb{R})$ such that for any $f \in \mathcal{U}$, the entropy map with respect to strong $C^{\infty}$-topology, is not locally constant at $f$.

    2. there exists $f \in \text{Diff}^{\infty}(\mathbb{R})$ such that the entropy map with respect to strong $C^{\infty}$-topology, is neither lower semi-continuous nor upper semi-continuous at $f$.

    Mathematics Subject Classification: 37B40, 37E05.


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