Mean dimension and its applications in dynamical systems - Jin; Lei (2023)
2 Study of the embedding for general group actions TOTAL We studied an embedding problem for amenable group G-actions: whether every minimal G-action can be embedded in the full G-shift on ([0, 1] d ) G . We answered this question in the negative by constructing a minimal G-action of mean dimension d/2 which cannot be embedded in the full G-shift on ([0, 1] d ) G for every infinite (countable discrete) amenable group G and every positive integer d. This example demonstrated that the optimal constant C stated in 2) will not exceed d/2. We therefore answered very partially Problem 4.5 stated in the proposal. This was written in the paper Mean dimension and a non-embeddable example for amenable groupactions, accepted for publication in the journal Fundamenta Mathematicae. 3 Study the generalization of the celebrated Mañé's theorem established in 1979 in the setting of continuum-wise expansive multiparameter actions with a view towards mean dimension TOTAL We studied directional mean dimension of Z k -actions (where k is a positive integer). On the one hand, we showed that there is a Z 2 -action whose directional mean dimension (considered as a [0, +?]-valued function on the torus) is not continuous. On the other hand, we proved that if a Z k -action is continuum-wise expansive, then it has finite directional mean dimension along any direction. This is a generalization (with a view towards Meyerovitch and Tsukamoto's theorem on mean dimension and expansive multiparameter actions) of a classical result due to Mañé: Any compact metrizable space admitting an expansive homeomorphism (with respect to a compatible metric) is finite dimensional. This was written in the paper Directional mean dimension and continuum-wise expansive Z^k -actions, accepted for publication in the journal Proceedings of the American Mathematical Society. Otro(s) aspecto(s) que Ud. considere importante(s) en la evaluación del cumplimiento de objetivos planteados en la propuesta original o en las modificaciones autorizadas por los Consejos. ...