Lyapunov exponents, periodic orbits, and horseshoes for semiflows on Hilbert spaces (Q2892813)

This very-well written paper provides a basic but important building block for the ergodic theory and global analysis of non-uniformly hyperbolic continuous-time dynamical systems (semi-flows) in infinite dimension. The main result is that for a semi-flow with an invariant measure, only one (trivial) zero Lyapunov exponent, and positive metric entropy then horseshoes are present as well as an abundance of hyperbolic periodic orbits. The key advance is the treatment of continuous-time dynamical systems and the results are new, even in a finite-dimensional set-up. The emphasis is also on dynamical systems arising possibly from dissipative parabolic partial differential equations, although the assumptions of the theorem are not proved for any concrete partial differential equation.