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A Lipschitz refinement of the Bebutov-Kakutani dynamical embedding theorem. (English) Zbl 1414.37005

A dynamical system on a compact metric space \(X\) is a group action of the additive group of real numbers. In the notation of this paper, a flow is a pair \((X,T)\) where \(X\) is compact metric space and \(T: \mathbb{R}\times X \to X, (x,t) \mapsto T_t x\) is a continuous action of \(\mathbb{R}\). For each flow there exists the rest set, denoted Fix\((X,T)\), consisting of points \(x\in X\) satisfying \(T_t x = x\) for all \(t\in \mathbb{R}\). Simultaneously \(\mathbb{R}\) acts on the space \(C(\mathbb{R})\) of maps \(\varphi: \mathbb{R}\to [0,1]\) endowed with the topology of uniform convergency. In this case \(\mathbb{R}\) acts continuously by translation: \(\mathbb{R}\times C(\mathbb{R}) \to C(\mathbb{R})\) sending \((s,\varphi(t) \to \varphi(t+s)\) and it is called shift flow or Bebutov flow. A Bebutov flow is a remarkable flow: there exist necessary and sufficient conditions to embed a compact flow \((X,T)\) into \((C(\mathbb{R}), \varphi)\). Presently, we say that a map \(f: X\to C(\mathbb{R})\) is an embedding of a flow \((X,T)\) if \(f\) is an \(\mathbb{R}\)-equivariant topological embedding.
The authors replace \(C(\mathbb{R})\) by its compact subset \(L(\mathbb{R})\subset C(\mathbb{R})\) which is the set of maps of \(\mathbb{R}\) into \([0,1]\) satisfying the one-Lipschitz condition \(|\varphi (s) - \varphi (t)| \leq |s - t|\). The obtained result is the following: a flow \((X,T)\) can be equivariantly embedded in \(L(\mathbb{R})\) if and only if Fix\((X,T)\) can be topologicaly embedded in \([0,1]\).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
54H20 Topological dynamics (MSC2010)
57N35 Embeddings and immersions in topological manifolds
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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References:

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