Shadowing for actions of some finitely generated groups

Abstract

We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag-Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally we show that any linear action of a non-abelian free group does not have the shadowing property.

Group Actions With Topologically Stable Measures

“…Since T g is expansive and g ∈ K, T K is expansive. As K is a normal subgroup of G by Proposition 2(N1) in [7], applying Lemma 3.5, we get the result.…”

“…Let n > 1 and assume that this lemma holds for each nilpotent group with nilpotent degree less than or equal to n − 1. Put G 1 = [G, G] and K =< G 1 , g >, then K has the nilpotent degree at most n − 1 by Proposition 2(N2) in [7]. It is known that G 1 is finitely generated by Lemma 6.8.4 in [5] and hence K is finitely generated.…”

“…al [7,9,10] introduced the notions of shadowing and invese shadowing for finitely generated group actions which are generalizations of those of topological stability, shadowing and inverse shadowing, respectively, for homeomorphisms on compact metric spaces.…”

“…In this paper, we consider a measure version of the dynamics of finitely generated group actions on compact metric spaces which were developed in [4,7,9,10]. More precisely, we introduce the notions of measure shadowing property and measure topological stability of group actions, and prove that if an action T of a finitely generated group G is measure expansive and has the measure shadowing property then it is measure topologically stable.…”

On Properties of Expansive Group Actions

“…Recently, in 2014, Osipov and Tikhomirov [12], introduced the notion of shadowing property for finitely generated group actions. Barzanouni in [3] used the notion of shadowing property to study chain recurrent sets whereas Chung and Lee in [5] showed that expansive action which has shadowing property are always topologically stable.…”