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To illustrate the more tractable properties of random dynamical systems, we consider a class of 2D maps with strong expansion on large—but non-invariant—subsets of their phase spaces. In the deterministic case, such maps are not precluded from having sinks, as derivative growth on disjoint time intervals can be cancelled when stable and unstable directions are reversed. Our main result is that when randomly perturbed, these maps possess positive Lyapunov exponents commensurate with the amount of expansion in the system. We show also that initial conditions converge exponentially fast to the stationary state, equivalently time correlations decay exponentially fast. These properties depend only on finite-time dynamics, and do not involve parameter selections, which are necessary for deterministic maps with nonuniform derivative growth. [ABSTRACT FROM AUTHOR]

We consider $$C^2$$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs. [ABSTRACT FROM AUTHOR]

We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions, including Lebesgue and Sinai–Reulle–Bowen (SRB) measures. Lower bounds do not hold in such a generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes. [ABSTRACT FROM PUBLISHER]

We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results extend Katok's work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes of periodically forced PDEs. [ABSTRACT FROM AUTHOR]