SRB Measures for A Class of Partially Hyperbolic Attractors in Hilbert spaces

TL;DR

This paper investigates the existence, uniqueness, and properties of SRB measures in infinite-dimensional Hilbert space dynamical systems with partially hyperbolic attractors, extending finite-dimensional results to infinite dimensions.

Contribution

It establishes conditions for the existence and finiteness of SRB measures in infinite-dimensional systems with hyperbolic attractors, including cases with topological mixing and H"older continuous splittings.

Findings

Existence of SRB measures for partially hyperbolic attractors with finite unstable directions.

Uniqueness and mixing properties of SRB measures for uniformly hyperbolic, topologically mixing systems.

Finiteness of SRB measures in non-wondering, uniformly hyperbolic systems with H"older continuous splittings.

Abstract

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is H\"older continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and and the splitting is H\"older continuous, then there exists at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set.