Mean dimension and Jaworski-type theorems

According to the celebrated Jaworski theorem, a finite-dimensional aperiodic dynamical system ( X , T ) embeds in the one-dimensional cubical shift ( [ 0 , 1 ] Z , shift ) . If X admits periodic points (still assuming dim ( X ) < ∞ ), then we show in this paper that periodic dimension perdim ( X , T ) < d / 2 implies that ( X , T ) embeds in the d-dimensional cubical shift ( ( [ 0 , 1 ] d ) Z , shift ) . This verifies a conjecture by Lindenstrauss and Tsukamoto for finite-dimensional systems. Moreover, for an infinite-dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in ( ( [ 0 , 1 ] d ) Z , shift ) . Furthermore, we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti–Crovisier tower theorem, to show that an extension ( X , T ) of an aperiodic finite-dimensional system whose mean dimension obeys mdim ( X , T ) < d / 16 embeds in the ( d + 1 ) -cubical shift ( ( [ 0 , 1 ] d + 1 ) Z , shift ) .