Self-adjoint Laplacians on partially and generalized hyperbolic attractors

TL;DR

This paper develops self-adjoint Laplacians and Markov semigroups on hyperbolic attractors with Gibbs measures, enabling the construction of associated diffusion processes, especially in geodesic flow contexts.

Contribution

It introduces a framework for defining self-adjoint Laplacians on complex hyperbolic attractors with singularities, extending classical leafwise Laplacians in new dynamical settings.

Findings

Existence of symmetric Markov semigroups on hyperbolic attractors.

Construction of associated Hunt diffusion processes when measure has full support.

Extension of classical leafwise Laplacians to new hyperbolic dynamical systems.

Abstract

We construct self-adjoint Laplacians and symmetric Markov semigroups on partially hyperbolic attractors and on hyperbolic attractors with singularities, endowed with Gibbs u-measures. If the measure has full support, we can also guarantee the existence of an associated symmetric Hunt diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on manifolds of negative sectional curvature the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians.