Shadowing in the case of nontransverse intersection

Abstract:

An example of a diffeomorphism of a three-dimensional manifold is constructed with the following properties: (i) this diffeomorphism satisfies Axiom A; (ii) there exist two hyperbolic fixed points $p_1$ and $p_2$ such that the unstable manifold $W^u(p_1)$ of $p_1$ and the stable manifold $W^s(p_2)$ of $p_2$ are one-dimensional and have nonempty intersection, $W^u(p_1)\cap W^s(p_2)\neq \varnothing$; and (iii) the diffeomorphism has the Hölder shadowing property with Hölder shadowing exponent $1/4$.