Absolute continuity of stable foliations for mappings of Banach spaces

TL;DR

This paper proves the absolute continuity of stable foliations for certain Banach space mappings, extending finite-dimensional hyperbolic theory results to infinite-dimensional PDE contexts, which is important for ergodic theory.

Contribution

It establishes absolute continuity of stable foliations in Banach spaces under conditions similar to dissipative PDEs, addressing geometric challenges absent in finite dimensions.

Findings

Proves absolute continuity for Banach space mappings

Extends hyperbolic theory to infinite-dimensional settings

Provides tools for ergodic theory in PDE contexts

Abstract

We prove the absolute continuity of stable foliations for mappings of Banach spaces satisfying conditions consistent with time-t maps of certain classes of dissipative PDEs. This property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space; it is also used in proofs of ergodicity. Absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory. On Banach spaces, the absence of nice geometric properties poses some additional difficulties.