Nonuniform hyperbolicity for C^1-generic diffeomorphisms

TL;DR

This paper investigates the ergodic properties of C^1-generic diffeomorphisms, demonstrating that such systems exhibit nonuniform hyperbolicity and many ergodic measures with supports on homoclinic classes and transitive sets, extending classical results.

Contribution

It proves that generic C^1 diffeomorphisms have ergodic measures supported on homoclinic classes and transitive sets that are nonuniformly hyperbolic, extending Pesin theory to C^1 settings.

Findings

Homoclinic classes support ergodic measures with full support.

Generic ergodic measures are nonuniformly hyperbolic with no zero Lyapunov exponents.

Hyperbolic measures with dominated Oseledets splittings satisfy Pesin's Stable Manifold Theorem.

Abstract

We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Man\'e in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C^1.