Multidimensional C⁰ transversality and the shadowing property for Axiom A diffeomorphisms

Petrov and Pilyugin (2015) generalized a notion of C⁰ transversality of Sakai (1995) using smooth curves. Their definition involves only continuous maps from Rⁿ to a manifold, which is a purely topological one. They also provided a sufficient condition for the C⁰ transversality in terms of homological nature. In this paper, we prove that such a homological condition of Axiom A diffeomorphisms is sufficient for enjoying the shadowing property. Moreover, it is proved that the C⁰ transversality of Axiom A diffeomorphisms with codimension one basic sets implies the homological condition.