Mean dimension, small entropy factors and an embedding theorem.

*(English)*Zbl 0978.54027The paper under review is an important contribution to the emerging theory of mean dimension of topological dynamical systems. Let \((X,T)\) be a topological dynamical system, where \(T\) is a self-homeomorphism of a compact space \(X\). The mean dimension of \((X,T)\) is a new invariant, suggested by Gromov and developed in [E. Lindenstrauss and B. Weiss, Isr. J. Math. 115, 1-24 (2000; Zbl 0978.54025), see above], aimed at distinguishing between dynamical systems with infinite entropy. Mean dimension is a variant of the classical concept of Lebesgue covering dimension, taking into account dynamics.

For a finite open cover \(\alpha\), denote by \(D(\alpha)\) the minimal order of a finite open cover \(\beta\) refining \(\alpha\). If in addition \(n\in\mathbb N\), denote \(\alpha^n=\alpha\vee T^{-1}\alpha\vee\ldots\vee T^{-n}\alpha\). Now the mean dimension of \((X,T)\) is defined by \[ \text{mdim}(X,T)=\sup_\alpha\lim_{n\to\infty}\tfrac 1n D(\alpha^n), \] where the supremum is taken over all finite open covers of \(X\). Note that if \(T=\text{Id}_X\) and the factor \(1/n\) is removed, then one obtains the Lebesgue covering dimension. The systems \((X,T)\) with zero mean dimension include (and unify) systems with finite entropy, finite-dimensional phase space, and those admitting at most countably many ergodic invariant measures. However, the main interest of the emerging theory lies namely with systems of positive – or even infinite – mean dimension. (Notice in this connection that for each \(t>0\) there exists a system \((X,T)\) with \(\text{mdim}(X,T)=t\).)

In a similar way, building on Bowen’s definition of the topological entropy of \((X,T)\), one can define the metric mean dimension of \((X,T)\). This invariant, \(\text{mdim}_M(X,T)\), is the infimum, taken over all compatible metrics \(d\) on \(X\), of the numbers \[ \text{mdim}_M(X,T,d):= \lim_{\varepsilon\to 0} \frac{\limsup_{n\to\infty}\frac 1n\log sp_n(\varepsilon)}{|\log \varepsilon|}, \] where \(sp_n(\varepsilon)\) is the minimal cardinality of a subset \(A\subseteq X\) with the property that for every \(x\in X\) there is an \(y\in A\) such that \(\max_{0\leq k<n} d(T^kx,T^ky)<\varepsilon\).

One of the first results of the theory asserts that always \(\text{mdim}(X,T)\leq\text{mdim}_M(X,T)\) (the paper by Lindenstrauss and Weiss [loc. cit.]), and that for systems admitting an infinite minimal factor the two dimensions coincide (the paper under review). The new concepts shed light on the important question of when is a system \((X,T)\) embeddable into a symbolic dynamical system of the form \((([0,1]^d)^{\mathbb Z},\sigma)\), where \(\sigma\) is a shift. Lindenstrauss and Weiss [loc. cit.] have previously shown that a necessary condition for the existence of such an embedding is that \(\text{mdim}(X,T)\leq d\). The present paper establishes the following result in the converse direction: if \((X,T)\) possesses an infinite minimal factor and \(\text{mdim}(X,T)<\frac 1{36}d\), then \((X,T)\) admits an embedding into the shift system \((([0,1]^d)^{\mathbb Z},\sigma)\). (Moreover, such embeddings are generic in a suitably defined sense.) These results can be considered as providing (an approximation to) a dynamical analogue of the classical Menger-Nöbeling embedding theorem for finite-dimensional compacta.

Recall that for a set \(E\subseteq X\) the orbit capacity of \(E\), denoted by \(\text{ocap}(E)\), as defined by M. Shub and B. Weiss [Ergodic Theory Dyn. Syst. 11, No. 3, 535-546 (1991; Zbl 0773.54011)], is the limit, as \(n\to\infty\), of the orders of families of sets formed by finite translates of \(E\) by the first \(n\) powers of \(T\). Subsets \(E\) with zero orbit capacity are said to have the small boundary property. Again, it was shown by Lindenstrauss and Weiss [loc. cit.] that if the topology of \(X\) has a basis formed by sets with the small boundary property, then \(\text{mdim}(X,T)=0\). The paper under review establishes the converse for those dynamical systems \((X,T)\) admitting infinite minimal factors. These results are a perfect dynamical analogue of Uryson’s classical result on the coincidence of the Lebesgue covering dimension and the small inductive dimension for compact metric spaces.

The paper also contains a number of other results, such as, for instance, a dichotomy between systems with zero mean dimension and those with \(\text{mdim}(X,T)>0\), the concept of the universal zero-mean dimensional factor of \(X\), etc.

For a finite open cover \(\alpha\), denote by \(D(\alpha)\) the minimal order of a finite open cover \(\beta\) refining \(\alpha\). If in addition \(n\in\mathbb N\), denote \(\alpha^n=\alpha\vee T^{-1}\alpha\vee\ldots\vee T^{-n}\alpha\). Now the mean dimension of \((X,T)\) is defined by \[ \text{mdim}(X,T)=\sup_\alpha\lim_{n\to\infty}\tfrac 1n D(\alpha^n), \] where the supremum is taken over all finite open covers of \(X\). Note that if \(T=\text{Id}_X\) and the factor \(1/n\) is removed, then one obtains the Lebesgue covering dimension. The systems \((X,T)\) with zero mean dimension include (and unify) systems with finite entropy, finite-dimensional phase space, and those admitting at most countably many ergodic invariant measures. However, the main interest of the emerging theory lies namely with systems of positive – or even infinite – mean dimension. (Notice in this connection that for each \(t>0\) there exists a system \((X,T)\) with \(\text{mdim}(X,T)=t\).)

In a similar way, building on Bowen’s definition of the topological entropy of \((X,T)\), one can define the metric mean dimension of \((X,T)\). This invariant, \(\text{mdim}_M(X,T)\), is the infimum, taken over all compatible metrics \(d\) on \(X\), of the numbers \[ \text{mdim}_M(X,T,d):= \lim_{\varepsilon\to 0} \frac{\limsup_{n\to\infty}\frac 1n\log sp_n(\varepsilon)}{|\log \varepsilon|}, \] where \(sp_n(\varepsilon)\) is the minimal cardinality of a subset \(A\subseteq X\) with the property that for every \(x\in X\) there is an \(y\in A\) such that \(\max_{0\leq k<n} d(T^kx,T^ky)<\varepsilon\).

One of the first results of the theory asserts that always \(\text{mdim}(X,T)\leq\text{mdim}_M(X,T)\) (the paper by Lindenstrauss and Weiss [loc. cit.]), and that for systems admitting an infinite minimal factor the two dimensions coincide (the paper under review). The new concepts shed light on the important question of when is a system \((X,T)\) embeddable into a symbolic dynamical system of the form \((([0,1]^d)^{\mathbb Z},\sigma)\), where \(\sigma\) is a shift. Lindenstrauss and Weiss [loc. cit.] have previously shown that a necessary condition for the existence of such an embedding is that \(\text{mdim}(X,T)\leq d\). The present paper establishes the following result in the converse direction: if \((X,T)\) possesses an infinite minimal factor and \(\text{mdim}(X,T)<\frac 1{36}d\), then \((X,T)\) admits an embedding into the shift system \((([0,1]^d)^{\mathbb Z},\sigma)\). (Moreover, such embeddings are generic in a suitably defined sense.) These results can be considered as providing (an approximation to) a dynamical analogue of the classical Menger-Nöbeling embedding theorem for finite-dimensional compacta.

Recall that for a set \(E\subseteq X\) the orbit capacity of \(E\), denoted by \(\text{ocap}(E)\), as defined by M. Shub and B. Weiss [Ergodic Theory Dyn. Syst. 11, No. 3, 535-546 (1991; Zbl 0773.54011)], is the limit, as \(n\to\infty\), of the orders of families of sets formed by finite translates of \(E\) by the first \(n\) powers of \(T\). Subsets \(E\) with zero orbit capacity are said to have the small boundary property. Again, it was shown by Lindenstrauss and Weiss [loc. cit.] that if the topology of \(X\) has a basis formed by sets with the small boundary property, then \(\text{mdim}(X,T)=0\). The paper under review establishes the converse for those dynamical systems \((X,T)\) admitting infinite minimal factors. These results are a perfect dynamical analogue of Uryson’s classical result on the coincidence of the Lebesgue covering dimension and the small inductive dimension for compact metric spaces.

The paper also contains a number of other results, such as, for instance, a dichotomy between systems with zero mean dimension and those with \(\text{mdim}(X,T)>0\), the concept of the universal zero-mean dimensional factor of \(X\), etc.

Reviewer: Vladimir Pestov (Wellington)

##### MSC:

54H20 | Topological dynamics (MSC2010) |

37B40 | Topological entropy |

37C45 | Dimension theory of smooth dynamical systems |

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\textit{E. Lindenstrauss}, Publ. Math., Inst. Hautes Étud. Sci. 89, 227--262 (1999; Zbl 0978.54027)

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##### References:

[1] | J. Auslander,Minimal flows and their extensions, Amsterdam, North-Holland (1988). · Zbl 0654.54027 |

[2] | W. Hurewicz andH. Wallman,Dimension Theory, Princeton University Press (1941). · JFM 67.1092.03 |

[3] | A. Jaworski,University of Maryland Ph.D. Thesis (1974). |

[4] | S. Kakutani, A proof of Bebutov’s theorem,J. of Differential Eq. 4 (1968), 194–201. · Zbl 0174.40101 |

[5] | E. Lindenstrauss, Lowering Topological Entropy,J. d’Analyse Math. 67 (1995), 231–267. · Zbl 0849.54031 |

[6] | E. Lindenstrauss andB. Weiss, On Mean Dimension, to appear inIsrael J. of Math. · Zbl 0978.54026 |

[7] | M. Shub andB. Weiss, Can one always lower topological entropy?,Ergod. Th. & Dynam. Sys. 11 (1991), 535–546. · Zbl 0773.54011 |

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