Abstract. According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system (X, T ) embeds in the 1-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

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a Z k -action on a compact metric space X, we study the following three problems closely related to mean dimension.(1) When is X isomorphic to the inverse limit of finite entropy systems? (2) Suppose the topological entropy h top (X) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer? (3) When can we embed X into the Z k -shift on the infinite dimensional cube ([0, 1] D ) Z k ? These were investigated for Z-actions in [Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. HautesÉtudes Sci. Publ. Math. 89 (1999) 227-262], but the generalization to Z k remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.

This paper forms the first part of a series by the authors [GMV16a, GMV16b] concerning the structure theory of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space X together with closed collections of cubes C n (X) ⊆ X 2 n , n = 1, 2, . . . satisfying some natural axioms.Antolín Camarena and Szegedy proved that from these axioms it follows that (certain) nilspaces are isomorphic (in a strong sense) to an inverse limit of nilmanifolds. The aim of our project is to provide a new self-contained treatment of this theory and give new applications to topological dynamics.This paper provides an introduction to the project from the point of view of applications to higher order Fourier analysis. We define and explain the basic definitions and constructions related to cubespaces and nilspaces and develop the weak structure theory, which is the first stage of the proof of the main structure theorem for nilspaces. Vaguely speaking, this asserts that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group.We also make some modest innovations and extensions to this theory. In particular, we consider a class of maps that we term fibrations, which are essentially equivalent to what are termed fibersurjective morphisms by Anatolín Camarena and Szegedy; and we formulate and prove a relative analogue of the weak structure theory alluded to above for these maps. These results find applications elsewhere in the project.