Rigidity for partially hyperbolic diffeomorphisms

Abstract

In this work we completely classify C ∞ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain C ∞ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and C ∞ center foliation.

A note on rigidity of Anosov diffeomorphisms of the three torus

“…So D f |E c f is uniform expanding and finally f is Anosov. From Corollary 1.2 and the above proposition we can conclude, by another approach, the result due to R.Varão, [17].…”

“…If f is derived from Anosov diffeomorphism and F c has the UBD property, it is possible to prove that f is Anosov. See [17].…”

Classification of partially hyperbolic diffeomorphisms under some rigid conditions

“…For perturbations of the linear Anosov map, the same result may also be obtained by using the first theorem in [SY19] once it is shown that the exponents of the Anosov diffeomorphism and its linear part are the same, which can be deduced from quasi-isometry of the foliations. A different approach to prove smooth conjugacy to a linear Anosov model was developed in [Var18] that uses smoothness of the center foliation plus extra requirements about the stable/unstable holonomies; to apply that approach one may have to establish that the hypotheses of our main theorem imply the requirements of [Var18], which does not seem to be direct. Another result related to the case that |λ c | = 1 is the one proved in [AVW15]: any partially hyperbolic diffeomorphism (that preserves a Liouville probability measure) close to the time-one map of a geodesic flow of a negatively curved surface with a smooth center foliation is the time-one map of a flow (close to the geodesic flow).…”

The disintegration of measures with periodic repetitive pattern

“…We may see the opposite of atomic disintegration when the conditional measures are equivalent to Lebesgue measures. This may also give important consequences for the dynamics ( [3,24,23] and references therein). It turns out that in dynamics it is not unusual to obtain either atomic or some "higher" regular disintegration (meaning for instance equivalent to Lebesgue) in many contexts ( [21,3,18,9]).…”