According to the Lindenstrauss-Tsukamoto Conjecture sufficient conditions for a topological dynamical system (X,T) to be embeddable in the d-cubical shift (([0,1]^d)^Z,shift) are that both the mean dimension and the periodic dimension of the system are strictly less than d/2. It is not hard to see that the conjecture is sharp. Recently the conjecture has been verified for the class of finite-dimensional systems (by the speaker) and for the class of extensions of aperiodic subshifts (joint work with Tsukamoto). In the talk I will present a proof for a class of systems admitting a finite-dimensional non-wandering set. This is the first non-trivial example of a class of systems both infinite-dimensional and non-aperiodic which is found to be embeddable in cubical shifts. The proof uses a concept of local markers which will be explained during the talk.