# On manifolds supporting distributionally uniquely ergodic diffeomorphisms

@article{Avila2012OnMS, title={On manifolds supporting distributionally uniquely ergodic diffeomorphisms}, author={Artur Avila and Bassam Fayad and Alejandro Kocsard}, journal={arXiv: Dynamical Systems}, year={2012} }

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this article we construct DUE diffeomorphisms supported on closed manifolds different from tori, providing some counterexamples to a conjecture proposed by Forni in [For08].

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## References

SHOWING 1-10 OF 14 REFERENCES

### Cohomological equations and invariant distributions for minimal circle diffeomorphisms

- Mathematics
- 2010

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As…

### Invariant distributions and time averages for horocycle flows

- Mathematics
- 2003

There are infinitely many obstructions to existence of smooth solutions of the cohomological equation Uu = f , where U is the vector field generating the horocycle flow on the unit tangent bundle SM…

### On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

- Mathematics
- 2012

Although invariant measures are a fundamental tool in Dynamical Systems, very little is known about distributions (i.e. linear functionals defined on some space of smooth functions on the underlying…

### On the cohomological equation for nilflows

- Mathematics
- 2005

Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there…

### On a class of homogeneous spaces of compact Lie groups

- Mathematics
- 1981

Let G be a connected compact Lie group, and U a connected closed subgroup of it. As is known [1 ], the difference r(G)-r(U) in the ranks is a homotopy invariant of the manifold G/U. When r(G) = r{U),…

### Representations of nilpotent Lie groups and their applications

- Mathematics
- 1989

Preface 1. Elementary theory of nilpotent Lie groups and Lie algebras 2. Kirillov theory 3. Parametrization of coadjoint orbits 4. Plancherel formula and related topics 5. Discrete cocompact…

### On the Greenfield-Wallach and Katok conjectures

- Mathematics
- 2007

We survey recent progress on the Greenfield-Wallach and Katok conjectures on globally hypoelliptic and cohomology free vector fields and derive a proof of the conjectures in dimension three. The…

### Basic theory and examples

- Mathematics
- 1990

Preface 1. Elementary theory of nilpotent Lie groups and Lie algebras 2. Kirillov theory 3. Parametrization of coadjoint orbits 4. Plancherel formula and related topics 5. Discrete cocompact…

### Compact Lie Groups

- Mathematics
- 2006

Compact Lie Groups.- Representations.- HarmoniC Analysis.- Lie Algebras.- Abelian Lie Subgroups and Structure.- Roots and Associated Structures.- Highest Weight Theory.

### Systèmes dynamiques topologiques I. Étude des limites de cobords

- Physics
- 1977

© Bulletin de la S. M. F., 1977, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord…