Implications of pseudo-orbit tracing property for continuous maps on compacta

Shadowing, entropy and minimal subsystems

“…We answer two more questions from [25]. First we show that an infinite transitive semi-distal system with shadowing must be a minimal equicontinuous system on a Cantor space, completing the results of [25,Theorem 5].…”

“…Proof It suffices to show that f is minimal, for then the required conclusion follows by Theorem 6 of [25]. The minimality of f follows either directly by Theorem 6.1, or in the following manner: if f is not minimal, then by considering a transitive point x and applying Theorem 6.2, we see that there exists (x, y) ∈ R( f × f ) ∩ Prox( f )\ , and in particular (x, y) ∈ Asy( f ) also, which contradicts the assumption that f is semi-distal.…”

“…We know by [25] that minimal points are dense in non-wandering dynamical systems having shadowing. If the system (X, f ) is also sensitive, we can improve the conclusion and show that both M( f )\ R R( f ) and R R( f )\ P( f ) are dense in X .…”

“…One of the reasons for considering non-wandering maps (e.g. instead of systems having the stronger property of transitivity) is that minimal points (and systems) are dense in every non-wandering system with shadowing [25], so it seems natural that knowledge about the structure of these systems can throw light on the possible types of global dynamical behaviors.…”

“…Our starting point (and main motivation) is [25] where many problems on dynamics of non-wandering systems with shadowing were left open. We answer most of the questions from [25].…”