Let $X \subset \mathbb{R}^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \mathbb{N}$ greater than the (Hausdorff) dimension of $X$ and assume that the set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a given Lipschitz map $h : X \to \mathbb{R}$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $\mu$. This is a probabilistic version of the Takens delay embedding theorem as proven by Sauer, Yorke and Casdagli. We also provide a non-dynamical probabilistic embedding theorem of similar type, which strengthens a previous result by Alberti, Bolcskei, De Lellis, Koliander and Riegler. In both cases, the key improvements compared to the non-probabilistic counterparts are the reduction of the number of required measurements from $2\dim X$ to $\dim X$ and using Hausdorff dimension instead of the box-counting one. We present examples showing how the use of the Hausdorff dimension improves the previously obtained results.

中文翻译：

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概率 Takens 定理

令 $X \subset \mathbb{R}^N$ 是一个 Borel 集，$\mu$ 是 $X$ 上的 Borel 概率测度，$T:X \to X$ 是 Lipschitz 和单射映射。修正 $k \in \mathbb{N}$ 大于 $X$ 的（Hausdorff）维数，并假设 $p$-周期点集的维数小于 $p$，因为 $p=1，\ldots， k-1$。我们证明，对于给定的 Lipschitz 映射 $h 的典型多项式扰动 $\tilde{h}$ ：X \to \mathbb{R}$，$k$-delay 坐标映射 $x \mapsto (\tilde{h }(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ 是一组全测度 $\mu$ 的单射。这是由 Sauer、Yorke 和 Casdagli 证明的 Takens 延迟嵌入定理的概率版本。我们还提供了一个类似类型的非动态概率嵌入定理，它加强了 Alberti、Bolcskei、De Lellis、Koliander 和 Riegler 先前的结果。在这两种情况下，与非概率对应物相比，关键的改进是将所需的测量数量从 $2\dim X$ 减少到 $\dim X$ 并使用 Hausdorff 维度而不是盒子计数维度。我们展示了如何使用 Hausdorff 维度改进先前获得的结果的示例。