Topological entropy of compact subsystems of transitive real line maps

Abstract

For a continuous map f from the real line (half-open interval [0, 1)) into itself let ent(f ) denote the supremum of topological entropies of f | K , where K runs over all compact f -invariant subsets of R ([0, 1), respectively). It is proved that if f is topologically transitive, then the best lower bound of ent(f ) is log √ 3 (log 3, respectively) and it is not attained. This solves a problem posed by Cánovas [Dyn. Syst. 24 (2009), no. 4, 473-483].

On almost specification and average shadowing properties

“…Note that, while in our setting the space X is compact, the definitions of the (almost) specification property and the two generalized shadowing properties remain meaningful without this assumption. But in a noncompact setting none of these properties (specification, almost specification, (asymptotic) average shadowing) is an invariant for topological conjugacy, as can be seen from the example in [19,Section 7] and Theorem 3.8 below. The reader can find several comments on the specification property and its relationship to the (asymptotic) average shadowing property in [14,15,16].…”

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

“…By passing to a subsequence we may assume that e k → e ∈ [0, 1] as k → ∞. From (23) we have that the sequence E(x, i k ) and the sequence with k-th term given by…”

“…If X is non-compact, then various properties we will be considering, such as the specification property and its variants, depend on ρ in a sense that changing the metric to an equivalent one may affect these properties. For an example where the specification property is lost with a change of metric, see [23,Proposition 7.1] (in other words, for non-compact spaces the specification property is no longer an invariant for topological conjugacy). For compact spaces the choice of metric is irrelevant.…”