Regularity of the Anosov Splitting and A New Description of the Margulis Measure

Citation

Hasselblatt, Boris (1989) Regularity of the Anosov Splitting and A New Description of the Margulis Measure. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/qkw4-xf63. https://resolver.caltech.edu/CaltechETD:etd-05232007-105707

Abstract

The Anosov splitting into stable and unstable manifolds of hyperbolic dynamical systems has been known to be Holder continuous always and differentiable under bunching or dimensionality conditions. It has been known, by virtue of a single example, that it is not always differentiable. High smoothness implies some rigidity in several settings.

In this work we show that the right bunching conditions can guarantee regularity of the Anosov splitting up to being differentiable with derivative of Holder exponent arbitrarily close to one. On the other hand we show that the bunching condition used is optimal. Instead of providing isolated examples we prove genericity of the low-regularity situation in the absence of bunching. This is the first time a local construction of low-regularity examples is provided.

Based on this technique we indicate how horospheric foliations of nonconstantly curved symmetric spaces can be made to be nondifferentiable by a smoothly small perturbation.

In the last chapter the Hamenstadt-description of the Margulis measure is rendered for Anosov flows and with a simplified argument. The Margulis measure arises as a Hausdorff measure for a natural distance on (un)stable leaves that is adapted to the dynamics.

Thesis Files