Self-adjoint Laplacians and symmetric diffusions on hyperbolic attractors

TL;DR

This paper develops self-adjoint Laplacians and symmetric diffusions on hyperbolic attractors with Gibbs u-measures, extending classical leafwise Laplacians and exploring properties like quasi-invariance and zero-energy functions.

Contribution

It constructs self-adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, including cases with full support measures and geodesic flows, advancing the understanding of diffusions in hyperbolic dynamics.

Findings

Existence of symmetric diffusions on hyperbolic attractors with Gibbs u-measures.

Self-adjoint extensions of classical leafwise Laplacians in specific dynamical systems.

Quasi-invariance of energy densities and zero-energy functions in certain cases.

Abstract

We construct self-adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs $u$-measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self-adjoint extensions of well-known classical leafwise Laplacians. We observe a quasi-invariance property of energy densities in the $u$-conformal case and the existence of nonconstant functions of zero energy.