Hyperbolic Dynamical Systems

Abstract

The theory of uniformly hyperbolic dynamical systems was initiated in the 1960's (though its roots stretch far back into the 19th century) by S. Smale, his students and collaborators, in the west, and D. Anosov, Ya. Sinai, V. Arnold, in the former Soviet Union. It came to encompass a detailed description of a large class of systems, often with very complex evolution. Moreover, it provided a very precise characterization of structurally stable dynamics, which was one of its original main goals. The early developments were motivated by the problem of characterizing structural stability of dynamical systems, a notion that had been introduced in the 1930's by A. Andronov and L. Pontryagin. Inspired by the pioneering work of M. Peixoto on circle maps and surface flows, Smale introduced a class of gradient-like systems, having a finite number of periodic orbits, which should be structurally stable and, moreover, should constitute the majority (an open and dense subset) of all dynamical systems. Stability and openness were eventually established, in the thesis of J. Palis. However, contemporary results of M. Levinson, based on previous work by M. Cartwright and J. Littlewood, provided examples of open subsets of dynamical systems all of which have an infinite number of periodic orbits.