A Lipschitz Refinement of the Bebutov–Kakutani Dynamical Embedding Theorem
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Abstract
We prove that an \(\mathbb {R}\)action on a compact metric space embeds equivariantly in the space of oneLipschitz functions \(\mathbb {R}\rightarrow [0,1]\) if its fixed point set can be topologically embedded in the unit interval. This is a refinement of the classical Bebutov–Kakutani theorem (1968).
Keywords
Compact universal flow Dynamical embedding Lipschitz functions Local section Bebutov–Kakutani theoremMathematics Subject Classification
37B05 54H201 Introduction
Theorem 1.1
(Bebutov–Kakutani) A flow (X, T) can be equivariantly embedded in \(C(\mathbb {R})\) if and only if \(\mathrm {Fix}(X,T)\) can be topologically embedded in the unit interval [0, 1].
The “only if” part is trivial because the set of fixed points of \(C(\mathbb {R})\) is homeomorphic to [0, 1]. So the main statement is the “if” part.
Although the Bebutov–Kakutani theorem is clearly a nice theorem, it has one drawback: The space \(C(\mathbb {R})\) is not compact (nor locally compact). So it is not a “flow” in the above definition. This poses the following problem:
Problem 1.2
Is there a compact invariant subset of \(C(\mathbb {R})\) satisfying the same universality?
Theorem 1.3
A flow (X, T) can be equivariantly embedded in \(L(\mathbb {R})\) if and only if \(\mathrm {Fix}(X,T)\) can be topologically embedded in the unit interval [0, 1].
As in the case of the Bebutov–Kakutani theorem, the “only if” part is trivial because the fixed point set \(\mathrm {Fix}(L(\mathbb {R}))\) is homeomorphic to [0, 1]. Since \(L(\mathbb {R})\) is compact, it is a more reasonable choice of such a “universal flow”.
The proof of Theorem 1.3 is based on the techniques originally used in the proof of the Bebutov–Kakutani theorem (in particular, the idea of local section). A main new ingredient is the topological argument given in Sect. 2, which has some combinatorial flavor.
Remark 1.4
2 Topological Preparations
Lemma 2.1
 (1)
\(\max _{x\in X} \left \left f(x)g(x)\right \right _\infty <\delta \).
 (2)
\(g(x)(0)=f(x)(0)\) and \(g(x)(a)=f(x)(a)\) for all \(x\in X\).
 (3)
\(g(X)\cap F_L[0,a] = \emptyset \).
Proof
 (1)
\(f(p_m)(a_n)  u_m(a_n) < \min \left( \delta /4, (1\tau )b/2\right) \) for all \(1\le m\le M\) and \(1\le n\le N\).
 (2)
\(u_m(a_{n+1})u_m(a_n) < \Delta \) for all \(1\le m\le M\) and \(1\le n\le N1\).
 (3)
The \((M+1)\) vectors \(e, u_1, \ldots , u_M\) are linearly independent.

\(g(x)(0) = f(x)(0)\) and \(g(x)(a)= f(x)(a)\).

\(g(x)(a_n) = \sum _{m=1}^M h_m(x) u_m(a_n)\) for \(1\le n\le N\).

We extend g(x) linearly. Namely, for \(t= (1\lambda ) a_n + \lambda a_{n+1}\) with \(0\le \lambda \le 1\) and \(0\le n\le N\) we set \(g(x)(t) = (1\lambda ) g(a_n) + \lambda g(a_{n+1})\).
Claim 2.2
\(g(x)\in L[0,a]\) and \(\left \left g(x)f(x)\right \right _\infty < \delta \).
Proof
Lemma 2.3
Proof
Lemma 2.4
Proof
Lemma 2.5
 (1)
\(\max _{x\in X} \left \left f(x)g(x)\right \right _\infty < \delta \).
 (2)
\(g(x)(0)=f(x)(0)\) and \(g(x)(a)=f(x)(a)\) for all \(x\in X\).
 (3)If \(x,y\in X\) and \(0\le \varepsilon \le a/2\) satisfythen \(\varepsilon =0\) and \(d(x,y) < \delta \).$$\begin{aligned} \forall t\in [0,a\varepsilon ]: g(x)(t+\varepsilon ) = g(y)(t) \end{aligned}$$
Proof
 (1)
\(f(p_m)(a_n)  u_m(a_n) < \min \left( \delta /4, (1\tau )b/2\right) \) for all \(1\le m\le M\) and \(1\le n\le N\).
 (2)
\(u_m(a_{n+1})u_m(a_n) < \Delta \) for all \(1\le m\le M\) and \(1\le n\le N1\).
 (3)
Define \(D_L u_m = (u_m(a_2)u_m(a_1), \ldots , u_m(a_{L+1})u_m(a_L))\in \mathbb {R}^L\). Then the \((M+1)\) vectors \(e, D_L u_1, \ldots , D_L u_M\) in \(\mathbb {R}^L\) are linearly independent.
 (4)For any \(\varepsilon > 0\) with \(\varepsilon + \Lambda \subset A\),$$\begin{aligned} u_1_{\Lambda },\> u_1_{\varepsilon +\Lambda },\> u_2_{\Lambda },\> u_2_{\varepsilon +\Lambda }, \ldots , u_m_{\Lambda }, \> u_m_{\varepsilon +\Lambda }~\text {are linearly independent in}~ \mathbb {R}^\Lambda \end{aligned}$$
3 Proof of Theorem 1.3
Lemma 3.1
The space \(C_{T,h}\left( X, L(\mathbb {R})\right) \) is not empty. Moreover for any \(\delta >0\) there exists \(f\in C_{T,h}\left( X, L(\mathbb {R})\right) \) satisfying \(\mathrm {Lip}(f)\le \delta \).
Proof
We borrow the next lemma from Auslander [1, p. 186, Corollary 6].
Lemma 3.2
Proof
Lemma 3.3
Proof
 (1)
\(g(x)(t)f_1(x)(t) < \delta \) for all \(x\in X\) and \(0\le t\le a\).
 (2)
\(g(x)(0)= f_1(x)(0)\) and \(g(x)(a)= f_1(x)(a)\) for all \(x\in X\).
 (3)
\(g(X) \cap F_L[0,a] = \emptyset \).
 For each \(s\in H(x)\), we set$$\begin{aligned} g_1(x)(t) = f_1(x)(t) + q(T_s x)\cdot u(T_s x)(ts) \quad \text {for}~ t\in [s,s+a]. \end{aligned}$$

For \(t\in \mathbb {R}{\setminus } \bigcup _{s\in H(x)} [s,s+a]\), we set \(g_1(x)(t) = f_1(x)(t)\).
Lemma 3.4
Proof
 (1)
\(g(x)(t) f_1(x)(t) < \delta \) for all \(x\in X\) and \(0\le t \le a\).
 (2)
\(g(x)(0)=f_1(x)(0)\) and \(g(x)(a)=f_1(x)(a)\) for all \(x\in X\).
 (3)If \(x,y\in X\) and \(0\le \varepsilon \le a/2\) satisfythen \(d(x,y) < \delta \).$$\begin{aligned} \forall t\in [0,a\varepsilon ]: g(x)(t+\varepsilon ) = g(y)(t) \end{aligned}$$
 For each \(s\in H(x)\), we set$$\begin{aligned} g_1(x)(t) = f_1(x)(t) + \tilde{q}(T_s x)\cdot u(T_s x)(ts) \quad \text {for}~t\in [s,s+a]. \end{aligned}$$

For \(t\in \mathbb {R}{\setminus } \bigcup _{s\in H(x)} [s,s+a]\), we set \(g_1(x)(t) = f_1(x)(t)\).

\(X{\setminus } F = \bigcup _{n=1}^\infty A_n\) and \((X{\setminus } F)\times (X{\setminus } F) {\setminus }\{(x,x):x\in X\} = \bigcup _{n=1}^\infty B_n\times C_n\).

\(G(A_n)\) are open and dense in the space \(C_{T,h}\left( X, L(\mathbb {R})\right) \) for all \(n\ge 1\).

\(G(B_n,C_n)\) are open and dense in the space \(C_{T,h}\left( X, L(\mathbb {R})\right) \) for all \(n\ge 1\).
Remark 3.5
The proof of the Bebutov–Kakutani theorem in [1, 4] used the idea of “constructing large derivative”. It is possible to prove Theorem 1.3 by adapting this idea to the setting of oneLipschitz functions. But this approach seems a bit tricky and less flexible than the proof given above. The above proof possibly has a wider applicability to different situations (e.g. other function spaces).
Notes
Acknowledgements
This paper was written when the third named author stayed in the Einstein Institute of Mathematics in the Hebrew University of Jerusalem. He would like to thank the institute for its hospitality.
References
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