Mean dimension and an embedding theorem for real flows. (English) Zbl 1471.37029
The notion of mean dimension was introduced by M. Gromov [Math. Phys. Anal. Geom. 2, No. 4, 323–415 (1999; Zbl 1160.37322)] as a topological invariant of dynamical systems to characterize systems with infinite topological entropies, and was systematically studied in [E. Lindenstrauss and B. Weiss, Isr. J. Math. 115, 1–24 (2000; Zbl 0978.54026); E. Lindenstrauss, Publ. Math., Inst. Hautes Étud. Sci. 89, 227–262 (1999; Zbl 0978.54027)].
In this paper, the authors introduce the notion of mean dimension for real flows, study its basic properties, and prove the following embedding theorem.
Theorem. Any real flow \((X, \mathbb R)\) of mean dimension strictly less than \(r\) admits an extension \((Y, \mathbb R)\) whose mean dimension is equal to the mean dimension of \((X, \mathbb R)\) and such that \((Y, \mathbb R)\) can be embedded in the \(\mathbb R\)-shift on the compact function space \(\{ f\in C(\mathbb R, [-1, 1]):\ \mathrm{supp}(\hat{f})\subset [-r, r]\), where \(\hat{f}\) is the Fourier transform of \(f\) considered as a tempered distribution.
In this paper, the authors introduce the notion of mean dimension for real flows, study its basic properties, and prove the following embedding theorem.
Theorem. Any real flow \((X, \mathbb R)\) of mean dimension strictly less than \(r\) admits an extension \((Y, \mathbb R)\) whose mean dimension is equal to the mean dimension of \((X, \mathbb R)\) and such that \((Y, \mathbb R)\) can be embedded in the \(\mathbb R\)-shift on the compact function space \(\{ f\in C(\mathbb R, [-1, 1]):\ \mathrm{supp}(\hat{f})\subset [-r, r]\), where \(\hat{f}\) is the Fourier transform of \(f\) considered as a tempered distribution.
Reviewer: Lin Shu (Beijing)
MSC:
| 37C45 | Dimension theory of smooth dynamical systems |
| 37B40 | Topological entropy |
| 37C10 | Dynamics induced by flows and semiflows |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37B02 | Dynamics in general topological spaces |
| 54C25 | Embedding |
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