N-expansive homeomorphisms with the shadowing property

We discuss the dynamics of n  -expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every n∈Nn∈N, we exhibit an n  -expansive homeomorphism, which is not (n−1)(n−1)-expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of n-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing n-expansive homeomorphisms that are not expansive.