On topological sequence entropy of piecewise monotonic mappings

Abstract

In this paper a. full classification of piecewise monotonic maps from the point of view of the topological sequence entropy is established.

Topological sequence entropy of interval maps

“…Moreover, it is possible to distinguish chaotic maps of type 2 ∞ from chaotic maps of type greater than 2 ∞ . Theorem 3 was proved previously in [3] for piecewise monotonic maps (abbreviated as pm-maps). Now, we are able to prove theorem 3 for any continuous interval map.…”

Computing explicitly topological sequence entropy: the unimodal case

“…Let us emphasize that the reader can find (sometimes implicit) proofs of all non-trivial results on maps from W(I) stated below (except of course those concerning sequence entropy) in [12]. [6] has been recently proved that hA ( f ) log 2 for any sequence A and any piecewise monotone map f E C(I)), which is similar to the well known fact that h( f ) log 2 for any unimodal map. Observe also that Card s(0, nz) = 2'~'2, so: [13], [18].…”

Sequence entropy for amenable group actions*

“…In 1974, Goodman [23] introduced the notion of topological sequence entropy in a way analogous to that of Kushnirenko, and studied some properties of topologically null systems. More details of topological sequence entropy for -actions can be found in [24]. In [25], Li first obtained a characterization of topological weak mixing by using topological sequence entropy.…”