We consider the attractor $\Lambda $ of a piecewise contracting map f defined on a compact interval. If f is injective, we show that it is possible to estimate the topological entropy of f (according to Bowen’s formula) and the Hausdorff dimension of $\Lambda $ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.

中文翻译：

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区间上的分段收缩图：豪斯多夫维数、熵和吸引子

我们考虑在紧区间上定义的分段收缩映射f的吸引子$\Lambda $ 。如果f是单射的，我们表明可以通过与系统轨道相关的复杂度来估计f的拓扑熵（根据鲍文公式）和$\Lambda $的豪斯多夫维数。具体来说，我们证明两个数字都为零。