2015/07/03 by C. A. Morales, Morales, C. A. · 1 citation
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary 54H20 #Secondary 49J53
paper · pdf · doi:10.48550/arxiv.1507.00989
openalex publication_date 2015/07/03 · openalex created_date 2022/08/29 · openalex updated_date 2026/07/28
We define shadowable points for homeomorphism on metric spaces. In the compact case we will prove the following results: The set of shadowable points is invariant, possibly nonempty or noncompact. A homeomorphism has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincides when every chain recurrent point is shadowable. Minimal or distal homeomorphisms of compact connected metric spaces have no shadowable points. The space is totally disconnected at every shadowable point for distal homeomorphisms (and conversely for equicontinuous homeomorphisms). A distal homeomorphism has the pseudo-orbit tracing property if and only if the space is totally disconnected (this improves Theorem 4 in \citemo).