The $C^{1+\alpha}$ hypothesis in Pesin theory revisited

TL;DR

This paper demonstrates that in the $C^{1+eta}$ setting for generic diffeomorphisms on 3-manifolds, there exist uncountably many hyperbolic ergodic measures with disjoint supports lacking stable and unstable manifolds, contrasting higher regularity cases.

Contribution

It revisits Pesin theory in the $C^{1+eta}$ context, showing the existence of measures with unusual properties and providing examples in dimension two without local genericity.

Findings

Existence of uncountably many hyperbolic ergodic measures with disjoint supports.

Supports lack stable and unstable manifolds in the $C^{1+eta}$ setting.

Contrasts with higher regularity where Pesin theory guarantees stable and unstable manifolds.

Abstract

We show that for every compact 3-manifold $M$ there exists an open subset of $\diff ^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures which are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher regularity case, where Pesin theory gives us the stable and the unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.