3,536 research outputs found
Homomorphisms between diffeomorphism groups
For r at least 3, p at least 2, we classify all actions of the groups
Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the
circle. This is the same as describing all nontrivial group homomorphisms
between groups of compactly supported diffeomorphisms on 1- manifolds. We show
that all such actions have an elementary form, which we call topologically
diagonal. As an application, we answer a question of Ghys in the 1-manifold
case: if M is any closed manifold, and Diff(M)_0 injects into the
diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the
answer is yes, even under more general conditions. Several lemmas on subgroups
of diffeomorphism groups are of independent interest, including results on
commuting subgroups and flows.Comment: Contains corrections and additional references. A revised version
will appear in Ergodic Theory and Dynamical System
A characterization of 3D steady Euler flows using commuting zero-flux homologies
We characterize, using commuting zero-flux homologies, those
volume-preserving vector fields on a -manifold that are steady solutions of
the Euler equations for some Riemannian metric. This result extends Sullivan's
homological characterization of geodesible flows in the volume-preserving case.
As an application, we show that the steady Euler flows cannot be constructed
using plugs (as in Wilson's or Kuperberg's constructions). Analogous results in
higher dimensions are also proved.Comment: 16 pages, we added proofs of analogous results in higher dimensions,
and a characterization of 3-dimensional Reeb field
Existence of common zeros for commuting vector fields on -manifolds
In E. Lima proved that commuting vector fields on surfaces with non-zero
Euler characteristic have common zeros. Such statement is empty in dimension
, since all the Euler characteristics vanish. Nevertheless,
\cite{Bonatti_analiticos} proposed a local version, replacing the Euler
characteristic by the Poincar\'e-Hopf index of a vector field in a region
, denoted by ; he asked:
\emph{Given commuting vector fields and a region where
, does contain a common zero of and
?}
\cite{Bonatti_analiticos} gave a positive answer in the case where and
are real analytic.
In this paper, we prove the existence of common zeros for commuting
vector fields , on a -manifold, in any region such that
, assuming that the set of collinearity of
and is contained in a smooth surface. This is a strong indication that the
results in \cite{Bonatti_analiticos} should hold for -vector fields.Comment: Final version, to appear in Annales de L'Institut Fourie
Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions
We show that most homogeneous Anosov actions of higher rank Abelian groups
are locally smoothly rigid (up to an automorphism). This result is the main
part in the proof of local smooth rigidity for two very different types of
algebraic actions of irreducible lattices in higher rank semisimple Lie groups:
(i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the
actions of cocompact lattices on Furstenberg boundaries, in particular,
projective spaces. The main new technical ingredient in the proofs is the use
of a proper "non-stationary" generalization of the classical theory of normal
forms for local contractions.Comment: 28 pages, LaTe
Strictly non-proportional geodesically equivalent metrics have
Suppose the Riemannian metrics and on a closed connected
manifold are geodesically equivalent and strictly non-proportional at
least at one point. Then the topological entropy of the geodesic flow of
vanishes.Comment: This is a slightly extended version of the paper submitted to ETDS.
16 pages, one .eps figur
Weak KAM for commuting Hamiltonians
For two commuting Tonelli Hamiltonians, we recover the commutation of the
Lax-Oleinik semi-groups, a result of Barles and Tourin ([BT01]), using a direct
geometrical method (Stoke's theorem). We also obtain a "generalization" of a
theorem of Maderna ([Mad02]). More precisely, we prove that if the phase space
is the cotangent of a compact manifold then the weak KAM solutions (or
viscosity solutions of the critical stationary Hamilton-Jacobi equation) for G
and for H are the same. As a corrolary we obtain the equality of the Aubry
sets, of the Peierls barrier and of flat parts of Mather's functions.
This is also related to works of Sorrentino ([Sor09]) and Bernard ([Ber07b]).Comment: 23 pages, accepted for publication in NonLinearity (january 29th
2010). Minor corrections, fifth part added on Mather's function (or
effective Hamiltonian
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