Shadowing and Expansivity in Sub-Spaces
  We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval, in which context some of our results can be extended. 
Andrew Barwell, Chris Good and Piotr Oprocha
math.DS