Representation of Zero-Dimensional Basis Sets of Surface Diffeomorphisms by Hyperbolic Homeomorphisms

V.Z. Grines and S.Kh. Aranson solved the classic J. Nielsen's problem of the realization of homotopy classes of homeomorphisms orientable surfaces with negative Euler characteristic, inducing hyperbolic action in the fundamental group in 1980. In each such homotopy class, a hyperbolic homeomorphism was constructed, the nonwandering set of which has a unique nontrivial zero-dimensional invariant set, which is the intersection of two geodesic laminations in the metric of constant negative curvature that carries the surface. The goal of this thesis is to distinguish such zero-dimensional basis sets of diffeomorphisms satisfying Axiom Smale's axiom, the restriction to which is topologically related to the dynamics of the restriction of a hyperbolic homeomorphism to an invariant nontrivial zero-dimensional set. The considered zero-dimensional basis sets are distinguished by the property of the spacious arrangement of stable and unstable manifolds of points of the basis set on a supporting surface. The concept of spacious location was introduced by R.V. Plykin in 1974 and was a generalization introduced earlier V.Z. Greenies concepts orientability of the base set. The presence of a spacious arrangement is manifested in the fact that the dynamics of restricting a diffeomorphism to such a set is closely related to the action induced by the surface diffeomorphism under study in its fundamental group. This becomes especially effective when the action of a diffeomorphism in the fundamental group is hyperbolic, that is, when any curve that is not homotopic to zero does not transform into a homotopic curve under any iterations of the diffeomorphism.