Fine structure of hyperbolic

The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms, by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeomorphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. In his remarkable article [23], Smale sets the foundations of the modern theory of dynamical systems. He defines the fundamental notion of hyperbolicity and relates it to structural stability. Let f be a smooth (at least C) diffeomorphism of a compact manifold M . A hyperbolic set for f is a closed f -invariant subset Λ ⊂ M such that the tangent bundle of the manifold over Λ splits as a direct sum of two subbundles that are invariant under the derivative, and the derivative of the iterates of the map expands exponentially one of the bundles (the unstable bundle) and contracts exponentially the stable subbundle. These bundles are in general only continuous, but they are integrable. Through each point x ∈ Λ, there exists a one-to-one immersed submanifold W (x), the stable manifold of x. This submanifold is tangent to the stable bundle at each point of intersection with Λ and is characterized by the fact that the orbit of each point y ∈ W (x) is asymptotic to the orbit of x, and, in fact, the distance between f(y) to f(x) converges to zero exponentially fast. These are the stable manifolds of Λ. They define a lamination whose holonomy is in general only Hölder continuous (a lamination is a partially defined foliation whose leaves are smooth but do not necessarily fill the whole manifold). However, in dimension two if the diffeomorphism is C + , i.e., C for some α > 0, the holonomy is also C + in the sense that the holonomy mapping between two local transversals extends to a C + local diffeomorphism. This property is crucial for the theory developed in the book under discussion. The same holds for the unstable manifolds which are the stable manifolds for the map f−1; see [9]. A basic set for f is a hyperbolic set Λ such that it is a maximal invariant subset of a neighborhood U, the periodic orbits are dense in Λ, and there is an orbit which is also dense in Λ. Such a basic set is persistent under perturbations of the diffeomorphism: there exists a neighborhood of N of f in the space of C diffeomorphisms of M endowed with the C topology, such that for each g ∈ N , the maximal invariant set Λ(g) in the neighborhood U is hyperbolic and there is a homemorphism h : Λ → Λ(g) that conjugates f with g. One of the major results in [23] is that if the nonwandering set Ω(f) is hyperbolic and the periodic points are dense in Ω(f), then the nonwandering set splits into a finite number of basic sets. Furthermore, if there is no cycle between basic sets of f (a cycle is a periodic sequence of basic sets such that the unstable manifold of each basic set intersects the stable manifold of the next basic set), then f is Ω-stable in the sense that there exists a neighborhood of f such that for each g in this neighborhood there exists a homeomorphism between Ω(f) and Ω(g) conjugating f and g.