[Paper Review] The C 1+ hypothesis in Pesin theory revisited

Christian Bonatti, Sylvain Crovisier+1 more|arXiv (Cornell University)|Jan 1, 2013

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper demonstrates that on every compact 3-manifold, a generic open set in the space of C¹ diffeomorphisms supports uncountably many disjoint, hyperbolic ergodic measures with attracting and repelling neighborhoods. Crucially, points in the supports of these measures lack stable and unstable manifolds—contrasting sharply with the classical Pesin theory, which guarantees such manifolds under higher regularity (C² or smoother).

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.

Motivation & Objective

• To investigate the behavior of ergodic measures in the C¹ topology on compact 3-manifolds.
• To determine whether stable and unstable manifolds exist for generic C¹ diffeomorphisms under the absence of higher regularity.
• To construct examples where the supports of hyperbolic measures are disjoint and admit both attracting and repelling neighborhoods.
• To contrast the C¹ case with the higher regularity case, where Pesin theory ensures the existence of stable and unstable manifolds.

Proposed method

• Use of genericity in the C¹ topology to identify open subsets of Diff¹(M) with desired dynamical properties.
• Construction of uncountably many ergodic probability measures that are hyperbolic and have disjoint supports.
• Demonstration that these measures' supports admit both attracting and repelling neighborhoods, implying no local stable or unstable manifolds.
• Application of topological and measure-theoretic techniques to analyze the structure of the supports and their dynamical behavior.
• Extension of results to dimension two, though without local genericity, to show broader applicability.
• Leveraging the failure of Pesin theory in C¹ to construct counterexamples to the existence of stable/unstable manifolds at typical points.

Experimental results

Research questions

• RQ1 Can uncountably many disjoint hyperbolic ergodic measures exist in the C¹ topology on a 3-manifold?
• RQ2 Do points in the supports of these measures lack stable and unstable manifolds despite being hyperbolic?
• RQ3 How does the absence of C² regularity affect the validity of classical Pesin theory in the C¹ setting?
• RQ4 Can such examples be constructed in dimension two, even without local genericity?
• RQ5 What topological and dynamical properties do the supports of these measures possess?

Key findings

• For every compact 3-manifold M, there exists an open subset of Diff¹(M) such that a generic diffeomorphism in this set admits uncountably many ergodic hyperbolic measures with disjoint supports.
• The supports of these measures admit both a basis of attracting neighborhoods and a basis of repelling neighborhoods, indicating no local stable or unstable dynamics.
• Points in the supports of these measures do not possess stable or unstable manifolds, contradicting the predictions of Pesin theory.
• The construction extends to dimension two, though without the local genericity present in the 3-dimensional case.
• The results highlight a fundamental breakdown of the classical stable manifold theory in the C¹ setting, even for hyperbolic measures.
• The failure of stable and unstable manifold existence at typical points underscores a critical distinction between C¹ and higher differentiability classes in smooth ergodic theory.