Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces (Q639265)

In this paper the authors extend the Katok theory of uniformly hyperbolic diffeomorphisms to the case of smooth not necessarily invertible maps of Hilbert spaces, which are non-uniformly hyperbolic with respect to ergodic Borel measures. In this interesting infinite-dimensional setting the authors show that if \(\mu\) has neither positive nor zero Lyapunov exponents, then \(\mu\) is supported on a stable periodic orbit. Also, if \(\mu\) is preserved by a smooth map \(f\) on Hilbert spaces, is ergodic and has no zero Lyapunov exponents, then either \(\mu\) is supported on a single periodic orbit (stable or unstable), or \(\mu\) is supported on the closure of a set of infinitely many unstable periodic orbits. Moreover, the authors show that if the measure-theoretic entropy of \(\mu\) is positive and if \(\mu\) has no zero Lyapunov exponents, then there exist both forward-invariant and bi-invariant horseshoes (defined by means of symbolic dynamics). These results can be applied to the case of systems defined by periodically driven nonlinear dissipative parabolic PDEs.