Abstract
We show that conservative partially hyperbolic diffeomorphism isotopic to the identity on Seifert 3-manifolds are ergodic.
Highlights
Since the foundational work of Grayson, Pugh, and Shub, a large focus of the study of partially hyperbolic dynamics has been to determine which of these systems are ergodic [19]
In the three-dimensional setting where each of the stable E s, unstable E u, and center E c bundles is one-dimensional, it is further conjectured that there is a unique obstruction to ergodicity: the presence of embedded tori tangent to the E u ⊕ E s direction [11]
We consider the family of hyperbolic orbifolds which consist of a sphere with exactly three cone points added. These orbifolds are small enough that their mapping class group is trivial, and using Theorem 1.1 we show that any partially hyperbolic diffeomorphism on the unit tangent bundle of such an orbifold is ergodic
Summary
Since the foundational work of Grayson, Pugh, and Shub, a large focus of the study of partially hyperbolic dynamics has been to determine which of these systems are ergodic [19]. In this setting of Seifert manifolds, Barthelmé, Fenley, Frankel, and Potrie have announced that any partially hyperbolic diffeomorphism isotopic to the identity is leaf conjugate to a topological Anosov flow [2]. Using this result would simplify some parts of our proof. We consider the family of hyperbolic orbifolds which consist of a sphere with exactly three cone points added These orbifolds are small enough that their mapping class group is trivial, and using Theorem 1.1 we show that any partially hyperbolic diffeomorphism on the unit tangent bundle of such an orbifold is ergodic. If f is volume preserving, it is accessible and ergodic
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