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Double variational principle for mean dimension

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Abstract

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

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Notes

  1. Throughout the paper we assume that the base of the logarithm is two. The natural logarithm (i.e. the logarithm of base e) is written as \(\ln (\cdot )\).

  2. \(f{:}\,\mathcal {X}\rightarrow (C_N)^\mathbb {Z}\) is called an embedding of a dynamical system if it is a topological embedding and satisfies \(f\circ T = \sigma \circ f\).

  3. The papers [GT, GQT] used the ideas of communication theory and signal processing. This is another manifestation of the intimate connections between mean dimension and information theory.

  4. It seems that these attract new interests of information theory researchers in the context of compressed sensing; see, e.g. [WV10, RJEP].

  5. The idea of introducing mean Hausdorff dimension was partly motivated by the study of Kawabata–Dembo [KD94, Proposition 3.2]. Roughly speaking, their result [KD94, Proposition 3.2] corresponds to Step 2.2 for \((\mathcal {X}, T) = (A^\mathbb {Z},\mathrm {shift})\) with \(A\subset \mathbb {R}^n\). In other words, Step 2.2 is a generalization of their result to arbitrary dynamical systems.

  6. There also exists a small issue about the tame growth of covering numbers condition. But we ignore it here.

  7. We always assume that the \(\sigma \)-algebra of a finite set is the largest one (the set of all subsets).

  8. We can show this by proving the data-processing inequality (Lemma 2.5) for the quantity defined by (2.4) in the case that \(\mathcal {X}\) and \(\mathcal {Y}\) are finite sets. See [CT06, Section 2.8].

  9. The continuity of \(\rho \) and \(\lambda \) is inessential. But we assume it for simplicity. Indeed in our applications, \(\mathcal {X}= \mathcal {Y}\), \(\rho \) is a distance function and \(\lambda \) is a constant.

  10. E.g. expanding signals in a wavelet basis, discarding small terms and quantizing the remaining terms.

  11. Although the “operational meaning” of rate distortion function is important for the understanding, we do not use it in the paper. So we do not give a complete explanation. See [LDN79, ECG94, Gra90] for the non-ergodic case.

  12. Indeed here we use only \(\underline{{\mathrm {mdim}}}_{\mathrm {H}}(\mathcal {X},T,d) < s\).

  13. An important point for us is that the statement is valid for each fixed \(\delta \) (not only the limits of \(\delta \rightarrow 0\)).

  14. Since we assume that A is a finite set, this just means that \(\mu _n(x)\rightarrow \mu (x)\) at each \(x\in A\).

  15. The coupling between X(k) and Y is given by the probability mass function

    $$\begin{aligned} \sum _{x'\in A^m} \pi _k(x,x') \mathbb {P}(Y=y|\mathcal {P}^m(X)=x'), \end{aligned}$$

    which converges to \(\mathbb {P}(\mathcal {P}^m(X)=x, Y=y)\).

  16. Indeed we cannot find this statement in [PS32]. The main theorem of their paper states that \(\dim \mathcal {X}\) is equal to the infimum of \(\underline{\dim }_{\mathrm {M}}(\mathcal {X},d)\) over \(d\in \mathscr {D}(\mathcal {X})\). But their argument actually proves Theorem 5.1.

  17. “Open” is easy. To show “dense”, take arbitrary \(f\in C(\mathcal {X},V)\) and \(\delta >0\). Choose \(0<\varepsilon <1/n\) such that \(d(x,y)<\varepsilon \) implies \(\left| \left| f(x)-f(y)\right| \right| <\delta \). There exists an \(\varepsilon \)-embedding \(\pi {:}\,\mathcal {X}\rightarrow P\) in a simplicial complex P of dimension \(\le \dim \mathcal {X}\). From Lemma 5.3 (2) and (3) in Section 5.2 we can find a linear embedding \(g:P\rightarrow V\) with \(\left| \left| g(\pi (x))-f(x)\right| \right| < \delta \). From Lemma 5.3 (1), \(\log \#(g(P), \left| \left| \cdot \right| \right| ,\varepsilon ')/\log (1/\varepsilon ')\) is less than \(\dim \mathcal {X} + 1/n\) for sufficiently small \(\varepsilon '\). This shows \(g\circ \pi \in A_n\). So \(A_n\) is dense. Therefore the main point of the proof of Theorem 5.2 is a “polyhedral approximation”. The basic idea of the proof of Theorem 5.1 is also a polyhedral approximation, but in a much more accurate way. See Section 5.3.

  18. Here is a technical point. The number \(A(P_n*Q_n)\) is defined by using the simplicial complex structure of \(P_n*Q_n\). We use the natural simplicial complex structure of the join \(P_n*Q_n\) here, not its subdivision introduced in (5.8).

  19. Since r is finite, this is more restricted than in the literatures [Sch95, LL18]. They consider automorphisms of general compact Abelian groups. But we study only the restricted class here for simplicity.

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Acknowledgements

This project was initiated at the Banff International Research Station meeting “Mean Dimension and Sofic Entropy Meet Dynamical Systems, Geometric Analysis and Information Theory” in 2017. We thank BIRS for hosting this workshop, and for providing ideal conditions for collaborations. We also thank the referee for her/his helpful comments.

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  1. Einstein Institute of Mathematics, Hebrew University, 91904, Jerusalem, Israel

    Elon Lindenstrauss

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Masaki Tsukamoto

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  1. Elon Lindenstrauss
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Correspondence to Masaki Tsukamoto.

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E.L. was partially supported by ISF Grant 891/15. M.T. was partially supported by JSPS KAKENHI 18K03275.

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Lindenstrauss, E., Tsukamoto, M. Double variational principle for mean dimension. Geom. Funct. Anal. 29, 1048–1109 (2019). https://doi.org/10.1007/s00039-019-00501-8

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