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This is a Quantitative Study study.

October 8, 2025Open Access

Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three

Key Points

Any conservative partially hyperbolic diffeomorphism is accessible in 3-manifolds, unless the fundamental group is virtually solvable.
These diffeomorphisms demonstrate ergodicity, confirming the Hertz-Hertz-Ures Ergodicity Conjecture.
The findings deepen our understanding of the dynamics of partially hyperbolic systems in three-dimensional spaces.
Ergodicity in this context implies that the system evolves in a way that is predictable over time, enhancing the structure of dynamical systems.

Abstract

We show that any conservative partially hyperbolic diffeomorphism homotopic to the identity is accessible unless the fundamental group of its ambient 3-manifold is virtually solvable. As a consequence, such diffeomorphisms are ergodic, giving an affirmative answer to the Hertz-Hertz-Ures Ergodicity Conjecture in the homotopy class of identity.

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Cite This Study

Feng et al. (Sat,) studied this question.

synapsesocial.com/papers/68e6d7971ffa7aa7d63d174b https://doi.org/https://doi.org/10.48550/arxiv.2506.00405