A representation of one-dimensional attractors of $A$-diffeomorphisms by hyperbolic homeomorphisms

We give a representation for the restrictions of $A$-diffeomorphisms of closed orientable surfaces of genus $>1$ from a homotopy class containing a pseudo-Anosov diffeomorphism to all one-dimensional attractors that do not contain special pairs of boundary periodic points. The representation is given by the restriction of a hyperbolic homeomorphism to an invariant zero-dimensional set formed by the intersection of two transversal geodesic laminations. It is shown how this result can be generalized to the representation of the restrictions of $A$-diffeomorphisms defined on a closed surface of any genus to arbitrary one-dimensional attractors.