Topology and topological sequence entropy

Abstract

Let X be a compact metric space and T : X −→ X be continuous. Let h * (T ) be the supremum of topological sequence entropies of T over all subsequences of Z + and S(X) be the set of the values h * (T ) for all continuous maps T on X. It is known that {0} ⊆ S(X) ⊆ {0, log 2, log 3, . . .} ∪ {∞}. Only three possibilities for S(X) have been observed so far, namely S(X) = {0}, S(X) = {0, log 2, ∞} and S(X) = {0, log 2, log 3, . . .} ∪ {∞}.In this paper we completely solve the problem of finding all possibilities for S(X) by showing that in fact for every set {0} ⊆ A ⊆ {0, log 2, log 3, . . .} ∪ {∞} there exists a one-dimensional continuum X A with S(X A ) = A. In the construction of X A we use Cook continua. This is apparently the first application of these very rigid continua in dynamics.We further show that the same result is true if one considers only homeomorphisms rather than continuous maps. The problem for group actions is also addressed. For some class of group actions (by homeomorphisms) we provide an analogous result, but in full generality this problem remains open. The result works also for an analogous class of semigroup actions (by continuous maps).

Uncountable Family of 0-Rigid Continua that are Homeomorphic to Their Inverse Limits

“…The rigidity of topological spaces has been studied in various ways by different authors, e.g., [1][2][3][4][5][6][7][8][10][11][12]. Cook continua are basic examples of non-degenerate continua that are rigid, and therefore, they play an important role in the study of rigid continua, also in continuum theory and dynamical systems in general.…”

“…Cook continua are basic examples of non-degenerate continua that are rigid, and therefore, they play an important role in the study of rigid continua, also in continuum theory and dynamical systems in general. There are many rigid continua or continua that are not 0-rigid that are constructed using Cook continua, see [2,6,10], where more references may be found. For example, in [2], Cook continua are used to construct stars, paths, and cycles of Cook continua.…”

Rigidity and Flexibility of Polynomial Entropy

“…In some spaces more can be said. For instance on the interval just three values 0, log 2 and ∞ can be obtained in this way [Can04]; see [SYZ20] for what is known in some other spaces. These rigidity results are accompanied by a flexibility result from [SYZ20] saying that for every set {0} ⊆ A ⊆ {0, log 2, log 3, .…”