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Abstract
We seek to extend to the setting of Banach space mappings a concept which has proven highly useful in the study of finite dimensional dissipative dynamical systems, that of SRB measures. These invariant measures, when they exist, govern the asymptotic dynamics of positive volume subsets of phase space: in short, SRB measures are to dissipative systems as Liouville measure is to conservative (e.g., Hamiltonian) systems. Our aim is to extend this theory to an infinite dimensional setting incorporating, for example, time-t solution mappings of partial differential equations with a suitably `dissipative' character, e.g., dissipative parabolic equations and dispersive wave equations.
The goal of this thesis is to generalize two results to this Banach space setting. The first result is a characterization of the SRB property in terms of the relationship between Lyapunov exponents, volume growth on unstable manifolds, and metric entropy. The second result says that an SRB measure is `visible' to a `positive volume' subset of phase space.
A complication arising for both results is that the finite dimensional theory heavily involves the notion of volume growth along unstable leaves, whereas Banach spaces do not possess an a priori notion of k-dimensional volume element. Another complication is that mappings in our infinite dimensional setting are far from the diffeomorphism or local diffeomorphism setting of the finite dimensional theory: our mappings may exhibit arbitrarily strong rates of contraction, and are not locally onto.