Cohomological equations and invariant distributions for minimal circle diffeomorphisms. (English) Zbl 1225.37052

Cohomological equations occur frequently, sometimes in disguised forms, in dynamical systems. When the dynamics are determined by a diffeomorphism \(f\) on a manifold, the most basic cohomological equation has the form of a first order linear difference equation \[ uf- u=\phi,\tag{\(*\)} \] where \(f: M\to M\) is a diffeomorphism, \(\phi: M\to \mathbb{R}\) is given and \(u\) is unknown. The authors assume that \(f\) and \(\phi\) are \(C^\infty\) and ask for smooth solutions \(u\). By analogy with the cohomology of groups, the function \(\phi\) can be thought of as a smooth cocycle over \(f\) and call it a (smooth) coboundary whenever \((*)\) has a \(C^\infty\) solution. This leads to a natural definition of the first cohomology space \(H^1(f,C^\infty(M))\).
An open question of some interest has been the computation of the \(C^\infty\) first cohomology space of an arbitrary minimal diffeomorphism.
The authors’ main result is the following:
If \(F: \mathbb{T}\to\mathbb{T}\) is an orientation-preserving diffeomorphism of the circle with irrational rotation number, and if \(\mu\) is its only invariant probability measure, then \(\mu\) is the only \(F\)-invariant distribution (up to multiplication by a real constant). A significant corollary is that a minimal \(C^\infty\) circle diffeomorphism is cohomologically \(C^\infty\)-stable if and only if its rotation number is Diophantine.
The authors note that their result is strictly one-dimensional. They provide relevant examples of smooth \(\mathbb{T}^2\) diffeomorphisms that are topologically conjugate to rigid rotations and exhibit higher-order invariant distributions in the upcoming paper titled “Invariant distributions for higher dimensional quasiperiodic maps”.


37E10 Dynamical systems involving maps of the circle
37C55 Periodic and quasi-periodic flows and diffeomorphisms
46F05 Topological linear spaces of test functions, distributions and ultradistributions
Full Text: DOI arXiv


[1] A. Avila and A. Kocsard, Invariant distributions for higher dimensional quasiperiodic maps , in preparation. · Zbl 1225.37052
[2] W. De Melo and S. Van Strien, One-dimensional Dynamics , Ergeb. Math. Grenzgeb. (3) 25 , Springer, Berlin, 1993. · Zbl 0791.58003
[3] é. Ghys, L’ invariant de Godbillon-Vey, Astérisque 177 -178 (1989), 155-181, Séminaire Bourbaki 1988/1989, no. 706. · Zbl 0707.57015
[4] -, Groups acting on the circle , Enseign. Math. (2) 47 (2001), 329-407. · Zbl 1044.37033
[5] -, Groups acting on the circle: A selection of open problems , conference lecture at “IIIéme cycle romand de mathématiques,” Les Diablerets, Switzerland, 2008, www.unige.ch/ tatiana/Diablerets08/ghys_diablerets.pdf.
[6] A. Haefliger and L. Banghe, “Currents on a circle invariant by a Fuchsian group”, in Geometric Dynamics (Rio de Janeiro, 1981) , Lecture Notes in Math. 1007 , Springer, Berlin, 1983, 369-378. · Zbl 0542.58004
[7] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers , 5th ed., Clarendon Press, New York, (1979). · Zbl 0423.10001
[8] M. C. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations , Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5-233. · Zbl 0448.58019 · doi:10.1007/BF02684798
[9] S. Hurder, “Dynamics and the Godbillon-Vey class: A history and survey” in Foliations: Geometry and Dynamics (Warsaw, 2000) , World Sci., River Edge, N. J., 2002, 29-60. · Zbl 1002.57058 · doi:10.1142/9789812778246_0003
[10] A. Katok, “Cocycles, cohomology and combinatorial constructions in ergodic theory” in Smooth Ergodic Theory and Its Applications (Seattle, Wash. 1999) , Proc. Sympos. Pure Math. 69 , Amer. Math. Soc., Providence, 2001, 107-173. · Zbl 0994.37003
[11] -, Combinatorial Constructions in Ergodic Theory and Dynamics , Univ. Lecture Ser. 30 , Amer. Math. Soc., Providence, 2003. · Zbl 1030.37001
[12] R. Langevin, “A list of questions about foliations” in Differential Topology, Foliations, and Group Actions (Rio de Janeiro, 1992) , Contemp. Math. 161 , Amer. Math. Soc., Providence, 1994, 59-80. · Zbl 0844.57028
[13] A. Navas, Actions de groupes de Kazhdan sur le cercle, Ann. Sci. Éc. Norm. Sup\(\acute{\mathrm e}\). (4) 35 (2002), 749-758. · Zbl 1028.58010 · doi:10.1016/S0012-9593(02)01107-2
[14] J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne , Ann. Sci. Éc. Norm. Sup\(\acute{\mathrm e}\). (4) 17 (1984), 333-359. · Zbl 0595.57027
[15] -, Centralisateurs et conjugaison différentiable des difféomorphismes du cercle , Astérisque 231 (1995), 89-242.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.