Shadowable points

ABSTRACT

A shadowable point is a point where the shadowing lemma holds for pseudo-orbits passing through it. We will prove the following results for homeomorphisms on compact metric spaces. The set of shadowable points is invariant, empty or non-empty, possibly non-compact. A homeomorphism has the pseudo-orbit tracing property (POTP) if and only if every point is shadowable. The chain recurrent and non-wandering sets coincide when every chain recurrent point is shadowable. The space is totally disconnected at every shadowable point for pointwise-recurrent homeomorphisms (and conversely for equicontinuous ones). In particular, pointwise-recurrent (like minimal or distal) homeomorphisms of non-degenerated continua have no shadowable points. The space admits a pointwise-recurrent homeomorphism with the POTP if and only if it is totally disconnected. A distal homeomorphism has the POTP if and only if the space is totally disconnected. We also exhibit homeomorphisms without the POTP for which the set of shadowable points is dense.

KEYWORDS:

• Shadowable point
• homeomorphism
• metric space

MCS/CCS/AMS Classification/CR Category numbers:

• (Primary) 37C50
• (Secondary) 37B45

Acknowledgments

This work is partially supported by MATHAMSUB 15 MATH05-ERGOPTIM, Ergodic Optimization of Lyapunov Exponents.

Disclosure statement

No potential conflict of interest was reported by the author.