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When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of p -periodic points admits a natural free action of ℤ / p ℤ for each prime number p. We are interested in the growth of its index and coindex as p → ∞. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, G -index, topological dynamics and marker property, preprint (2020), arXiv:2012.15372. [ABSTRACT FROM AUTHOR]

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps \pi : (X, T) \to (Y, S) between dynamical systems and assume that the mean dimension of the domain (X, T) is larger than the mean dimension of the target (Y, S). We exhibit several situations for which the maps \pi necessarily have positive conditional metric mean dimension. This study has interesting consequences to the theory of topological conditional entropy. In particular it sheds new light on a celebrated result of Lindenstrauss and Weiss [Israel J. Math. 115 (2000), pp. 1–24] about minimal dynamical systems non-embeddable in [0,1]^{\mathbb {Z}}. [ABSTRACT FROM AUTHOR]

We study the problem of embedding minimal dynamical systems into the shift action on the Hilbert cube [ 0 , 1 ] N Z . This problem is intimately related to the theory of mean dimension, which counts the average number of parameters for describing a dynamical system. Lindenstrauss proved that minimal systems of mean dimension less than cN for c = 1 / 36 can be embedded in [ 0 , 1 ] N Z , and asked what is the optimal value for c. We solve this problem by showing embedding is possible when c = 1 / 2 . The value c = 1 / 2 is optimal. The proof exhibits a new interaction between harmonic analysis and dynamical coding techniques. [ABSTRACT FROM AUTHOR]

We study the problem of embedding arbitrary Z k -actions into the shift action on the infinite dimensional cube [ 0 , 1 ] D Z k . We prove that if a Z k -action X satisfies the marker property (in particular if X is a minimal system without periodic points) and if its mean dimension is smaller than D / 2 then we can embed it in the shift on [ 0 , 1 ] D Z k . The value D / 2 here is optimal. The proof goes through signal analysis. We develop the theory of encoding Z k -actions into band-limited signals and apply it to proving the above statement. Main technical difficulties come from higher dimensional phenomena in signal analysis. We overcome them by exploring analytic techniques tailored to our dynamical settings. The most important new idea is to encode the information of a tiling of R k into a band-limited function which is constructed from another tiling. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to point out a new connection between information theory and dynamical systems. In the information theory side, we consider rate distortion theory, which studies lossy data compression of stochastic processes under distortion constraints. In the dynamical systems side, we consider mean dimension theory, which studies how many parameters per iterate we need to describe a dynamical system. The main results are new variational principles connecting rate distortion function to metric mean dimension. [ABSTRACT FROM PUBLISHER]

Mean dimension is a topological invariant of dynamical systems counting the number of parameters averaged by dynamics. Brody curves are Lipschitz holomorphic maps f:C→CPN, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. As a key novel ingredient, we use an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss. [ABSTRACT FROM AUTHOR]

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $${\mathbb{Z}^k}$$ -action on a compact metric space X, we study the following three problems closely related to mean dimension. These were investigated for $${\mathbb{Z}}$$ -actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227-262, 1999), but the generalization to $${\mathbb{Z}^k}$$ remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3). A key ingredient is a new method to continuously partition every orbit into good pieces. [ABSTRACT FROM AUTHOR]

This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of topological pressure. We consider a minimax problem for the sum of rate distortion dimension and the integral of a potential function. We prove that the minimax value is equal to the mean dimension with potential for a dynamical system having the marker property. The basic idea of the proof is a dynamicalization of geometric measure theory. [ABSTRACT FROM AUTHOR]