A panorama of specification-like properties and their consequences

Abstract

We oer an overview of the specification property, its relatives and their consequences. We examine relations between specification-like prop- erties and such notions as: mixing, entropy, the structure of the simplex of invariant measures, and various types of the shadowing property. We pay special attention to these connections in the context of symbolic dynamics. The specification property is the ability to find a single point following -close an arbitrary collection of orbit segments, provided that the tracing point is allowed to spend a fixed (for given ) time between consecutive segments. Rufus Bowen introduced the specification property in his seminal paper of 1971 (15) on Axiom A dieomorphisms. In recent years this notion and its generalizations served as a basis for many developments in the theory of dynamical systems. This property is closely related to the study of hyperbolic systems initiated during the 1960's. Around that time Stephen Smale noticed that certain maps arising from forced oscillations and geodesic flows on surfaces of negative curvature had similar geometric and analytic properties. This motivated his definition of what we know today as uniformly hyperbolic systems. At the same time, the Russian school (an incomplete list contains such names as Anosov, Sinai, Katok) worked intensively on Anosov systems, that is, dieomorphisms of manifolds under which