Amenable upper mean dimensions. (English) Zbl 1480.37038
This paper is devoted to the notion of upper measure-theoretic mean dimension and to the variational principle of upper metric mean dimensions.
It is known that the entropy is the most important invariant to measure the complexity of dynamical systems. Furthermore, it plays an essential role in information theory, dimension theory and fractal geometry. A part of this paper is devoted to the concept of mean dimension, its applications, and variational principles. The main goal of this work is a formulation of upper metric mean dimension and upper measure-theoretic mean dimension for amenable group actions. A relation between upper metric mean dimensions and pseudo-orbits is also discussed.
It is known that the entropy is the most important invariant to measure the complexity of dynamical systems. Furthermore, it plays an essential role in information theory, dimension theory and fractal geometry. A part of this paper is devoted to the concept of mean dimension, its applications, and variational principles. The main goal of this work is a formulation of upper metric mean dimension and upper measure-theoretic mean dimension for amenable group actions. A relation between upper metric mean dimensions and pseudo-orbits is also discussed.
Reviewer: Symon Serbenyuk (Kyïv)
MSC:
| 37C45 | Dimension theory of smooth dynamical systems |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 28D20 | Entropy and other invariants |
| 37B65 | Approximate trajectories, pseudotrajectories, shadowing and related notions for topological dynamical systems |
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