Shadowing is generic

Shadowing is generic on dendrites

“…Yano showed that shadowing is generic for homeomorphisms of the unit circle [12], and Odani proved that shadowing is generic for homeomorphisms on smooth manifolds with dimension at most three [10]. Pilyugin and Plamenevska extended this to homeomorphisms on compact manifolds without boundary but with a handle decomposition [11].…”

On the genericity of the shadowing property for conservative homeomorphisms

“…S. Y. Pilyugin and O. B. Plamenevskaya were able to improve this result in [15] to any dimension in the case of smooth manifolds. In 2005, P. Koscielniak established in [9] the genericity of the shadowing property for homeomorphisms on a compact manifold which possesses a triangulation (smooth manifolds or topological manifolds of dimension ≤ 3 for example) or a handle decomposition (smooth manifolds or manifolds of dimension ≥ 6 for example).…”

Generic Homeomorphisms with Shadowing of One-Dimensional Continua

“…Recall that a set is G δ if it is a countable intersection of open sets and it is residual if it contains a dense G δ subset. For instance, it is known that the shadowing property is generic for X a compact manifold [6,Theorem 1] or a Cantor set [3,Theorem 4.3]. Recall that f ∈ H(X) has the shadowing property if for all ε > 0 there is δ > 0 such that if {x i } i∈Z is a δ-pseudo orbit then there is y ∈ X such that dist(f i (y), x i ) < ε for all i ∈ Z.…”