In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension > 1 in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

中文翻译：

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分岔、对称和固定子空间的概念

在承认对称性的平稳分岔问题的背景下，本文着重于固定子空间（FS）的关键概念，并回顾了一些旨在检测各种情况下的分岔解决方案的应用。我们开始回顾其常用的简化版本，旧的等变分岔引理 (EBL)，其中 FS 是一维的；然后我们在非半简单临界特征值的典型情况下提供第一个推广，其中对称性的存在产生了非平凡的情况。接下来，我们考虑 FSs 的情况 > 1在非常不同的情况下。首先，依靠拓扑索引理论，特别是 Krasnosel'skii 定理，我们提供了一个在很大程度上适用于 EBL 扩展的陈述。其次，我们提出了一个完全不同的新应用，它将对称特性与分叉解的稳定性概念相结合。我们还提供了一些简单的例子，构造特设来说明各种情况。