No-shadowing for singular hyperbolic sets with a singularity
Xiao Wen, Lan Wen
Abstract
We prove that every singular hyperbolic chain transitive set with a singularity does not admit the shadowing property. Using this result we show that if a star flow has the shadowing property on its chain recurrent set then it satisfies Axiom A and the no-cycle conditions; and that if a multisingular hyperbolic set has the shadowing property then it is hyperbolic.
Topics & Concepts
SingularityTransitive relationMathematicsProperty (philosophy)Pure mathematicsHyperbolic setSet (abstract data type)Mathematical analysisComputer scienceCombinatoricsEpistemologyPhilosophyProgramming languageMathematical Dynamics and FractalsQuantum chaos and dynamical systemsGeometric and Algebraic Topology