Self-adjoint Laplacians and Symmetric Diffusions on Hyperbolic Attractors

In this thesis analysis on the attractors of hyperbolic dynamical systems is established in terms of Dirichlet forms. We construct self-adjoint Laplacians and symmetric Markov semigroups on uniformly, partially and generalized hyperbolic attractors, as well as on attractors with nonuniformly hyperbolic structure, endowed with SRB measures or Gibbs u-measures. If the measure has full support, we also guarantee the existence of an associated symmetric Hunt diffusion process. 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 nonconstant harmonic functions of zero energy in the ergodic case. 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.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2022

Author dc:creator

• Alikhanloo, Shayan

Repository record source_url

https://pub.uni-bielefeld.de/record/2961374

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2961374