LYAPUNOV EXPONENTS, PERIODIC ORBITS, AND HORSESHOES FOR SEMIFLOWS ON HILBERT SPACES

摘要

This work can be seen as a small step in a program to build an ergodic theory for infinite dimensional dynamical systems, a theory the domain of applicability of which will include systems defined by evolutionary PDEs. To reduce the scope, we focus on the ergodic theory of chaotic systems, on nonuniform hyperbolic theory, to he even more specific. In finite dimensions, a basic nonuniform hyperbolic theory already exists (see e.g.[8] [9], [11], [10], [2] and [4]). This body of results taken together provides a fairly good foundation for understanding chaotic phenomena on a qualitative, theore+ical level. While an infinite dimensional theory is likely to be richer and more complex, there is no reason to reinvent all materjal from scratch. It is thus logical to start by determining which parts of finite dimensional hyperbolic theory can be extended to infinite dimensions. Our paper is an early step (though nut the first step) in this effort. With an eye toward applications to systems defined by PDEs, emphasis will be given to continuous-time systems or semifiows. Furthermore, it is natural to first consider settings compatible with dissipative parabolic PDEs, for these systems have a finite dimensional flavor (see e.g. [14], [13], and [1])