We study the dynamics of the group of isometries of $L_p$-spaces. In particular, we study the canonical actions of these groups on the space of $\delta$-isometric embeddings of finite dimensional subspaces of $L_p(0,1)$ into itself, and we show that for $p \neq 4,6,8,\ldots$ they are $\varepsilon$-transitive provided that $\delta$ is small enough. We achieve this by extending the classical equimeasurability principle of Plotkin and Rudin. We define the central notion of a Fraisse Banach space which underlies these results and of which the known separable examples are the spaces $L_p(0,1)$, $p \neq 4,6,8,\ldots$ and the Gurarij space. We also give a proof of the Ramsey property of the classes $\{\ell_p^n\}_n$, $p\neq 2,\infty$, viewing it as a multidimensional Borsuk-Ulam statement. We relate this to an arithmetic version of the Dual Ramsey Theorem of Graham and Rothschild as well as to the notion of a spreading vector of Matousek and Rodl. Finally, we give a version of the Kechris-Pestov-Todorcevic correspondence that links the dynamics of the group of isometries of an approximately ultrahomogeneous space $X$ with a Ramsey property of the collection of finite dimensional subspaces of $X$.

中文翻译：

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L 空间的合并和 Ramsey 性质

我们研究了 $L_p$-空间等距组的动力学。特别地，我们研究了这些群在 $L_p(0,1)$ 的有限维子空间的 $\delta$-等距嵌入空间上的规范行为，并且我们证明对于 $p\neq 4,6 ,8,\ldots$ 如果 $\delta$ 足够小，它们是 $\varepsilon$-transitive。我们通过扩展 Plotkin 和 Rudin 的经典等量性原则来实现这一点。我们定义了 Fraisse Banach 空间的中心概念，它是这些结果的基础，其中已知的可分离示例是空间 $L_p(0,1)$, $p \neq 4,6,8,\ldots$ 和 Gurarij 空间. 我们还给出了类 $\{\ell_p^n\}_n$, $p\neq 2,\infty$ 的 Ramsey 性质的证明，将其视为多维 Borsuk-Ulam 语句。我们将此与 Graham 和 Rothschild 的对偶拉姆齐定理的算术版本以及 Matousek 和 Rodl 的传播向量的概念联系起来。最后，我们给出了 Kechris-Pestov-Todorcevic 对应关系的一个版本，它将近似超齐次空间 $X$ 的等距组的动力学与 $X$ 的有限维子空间集合的 Ramsey 性质联系起来。