Minimal sets: An introduction to topological dynamics

Commutativity and non-commutativity of topological sequence entropy

“…For these maps a kind of "symbolic dynamics" was introduced which could be seen as an extension of a standard tool in one-dimensional dynamics, "the adding machine" [7] (in our case we could better speak of a "substracting machine" as we will see later). Namely, in [8] was showed that for any w-map / it is possible construct a family {^a(/)}aeZ°° ( or simply {^a}aeZ 00 once there is no ambiguity on /) of pairwise disjoint (possibly degenerate) compact subintervals of [0,1] satisfying the key properties (Pl)-(P4) described below.…”

On low dimensional minimal sets

“…The Alexander-Spanier cohomology theory is used here. Using this result, Chu has answered questions that were raised by Gottschalk [6]. He proved that the universal curve of Menger and the universal curve of Sierpinski are not minimal sets under any connected topological group.…”

Left almost periodicity does not imply right almost periodicity

“…In this note, we shall show that a left almost periodic function is not necessarily right almost periodic even if the group G is a Lie group. This answers a problem in [3]. For the notions of almost periodic functions, we refer to [l], [4].…”

“…A real valued continuous function ƒ, defined on G, is left {right} almost periodic iff for any e>0, there is a left {right} syndetic subset 3 A of G such that \f{ax) -f(x)\ <e for all aÇzA, #£G. In this note, we shall show that a left almost periodic function is not necessarily right almost periodic even if the group G is a Lie group.…”