  According to a conjecture of Lindenstrauss and Tsukamoto, a topological
dynamical system $(X,T)$ is embeddable in the $d$-cubical shift
$(([0,1]^{d})^{\mathbb{Z}},\ shift)$ if both its mean dimension and periodic
dimension are strictly bounded by $\frac{d}{2}$. We verify the conjecture for
the class of systems admitting finite dimensional non-wandering sets (under the
additional assumption of closed periodic points set). The main tool in the
proof is the new concept of local markers. Continuing the investigation of
(global) markers initiated in previous work it is shown that the marker
property is equivalent to a topological version of the Rokhlin Lemma. Moreover
new classes of systems are found to have the marker property, in particular,
extensions of aperiodic systems with a countable number of minimal subsystems.
Extending work of Lindenstrauss we show that for systems with the marker
property vanishing mean dimension is equivalent to the small boundary property.
Finally we answer a question by Downarowicz in the affirmative: the small
boundary property is equivalent to admitting a zero-dimensional isomorphic
extension.
