Mean Dimension and Metric Mean Dimension for Non-autonomous Dynamical Systems

Sofic Mean Dimension

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-LindenstraussWeiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.

From rate distortion theory to metric mean dimension: variational principle

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 second we need to describ...

Hausdorff Dimension and mean porosity

Mean Dimension, Mean Rank, and Von Neumann-lück Rank

We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group Γ, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced Γ-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück rank of M all coincide. As applications, we establish an addition formula for mean dimension o...

Deformation of Brody Curves and Mean Dimension

The main purpose of this paper is to show that ideas of deformation theory can be applied to “infinite dimensional geometry”. We develop the deformation theory of Brody curves. Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite dimensional. As an application we pr...