DOI: 10.2478/s12175-007-0057-9 Math. Slovaca 58 (2008), No. 1, 95–100 ON SHADOWING PROPERTY FOR INVERSE LIMIT SPACES Ekta Shah — T. K. Das (Communicated by Michal Feˇ ckan ) ABSTRACT. We study here the G-shadowing property of the shift map σ on the inverse limit space X f , generated by an equivariant self-map f on a metric G-space X. c 2008 Mathematical Institute Slovak Academy of Sciences Introduction The theory of shadowing is a significant part of the qualitative theory of discrete dynamical systems. From numerical point of view, if a dynamical system has the shadowing property, then numerically obtained orbits reflect the real behaviour of trajectories of the systems. Furthermore, we may consider the shadowing property as a weak form of stability of dynamical systems with respect to C 0 perturbations. The present paper concerns the shadowing property for shift maps on inverse limit spaces. A sequence {x n : n ≥ 0} is called a δ-pseudo orbit of a self continuous map f on X if d(f (x n ),x n+1 ) <δ, for all n ≥ 0. Map f is said to have the shadowing property if for a given ǫ> 0, there is a δ> 0 such that for every δ-pseudo orbit {x n : n ≥ 0} there is a point x in X satisfying d(f n (x),x n ) <ǫ, for all n ≥ 0. For a compact metric space X, X Z is the compact metric space of all two-sided sequences (x n ) n∈Z , endowed with the product topology. Let f be a continuous self-map on X. Then the closed subspace X f =  (x n ): f (x n )= x n+1 for all n ∈ Z  of X Z together with the associated shift map σ : X f → X f defined by σ((x n )) = (y n ), where y n = x n+1 for all n ∈ Z, is the inverse limit space of f . By a metric G-space X we mean a metric space X on which a topological group G acts continuously by an action θ. For x in X, g in G we denote θ(g,x) 2000 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 54C10, 37C50; Secondary 54H20. Keywords: G-space, shadowing property, inverse limit space.