Self-adjoint Laplacians and symmetric diffusions on hyperbolic attractors

Abstract

We prove the existence of symmetric diffusions and self-adjoint Laplacians on (uniformly) hyperbolic attractors, endowed with SRB-measures. The proof is based on Dirichlet form theory. We observe some features of such diffusions, for instance, a quasi-invariance property of energy densities in the u-conformal case and the existence of non-constant harmonic functions of zero energy in the ergodic case. Contents 24 Appendix A. Disintegration and Rokhlin's theorem 26 Appendix B. The geometric construction of SRB-measures 26 Appendix C. Dirichlet integrals on weighted manifolds 27 Appendix D. Superposition of closable forms 30 References 31