A shadowable chain recurrent set with an attached hyperbolic singularity

We prove that every factor map between topological flows preserves the standard shadowing property if it is injective except for a closed orbit that shrinks to a singularity. As an application, we construct a C ∞ -flow on a four-dimensional sphere whose nonwandering set contains an attached hyperbolic singularity yet possesses the standard shadowing property. This gives a counterexample to a conjecture given by Arbieto et al. (2024) [1] .