With Sylvain Crovisier and Todd Fisher, we have just finished the paper Local perturbations of conservative C1-diffeomorphisms. It is now available on arxiv.
While studying the entropy of diffeomorphisms in this paper, we needed a number of lemmas for perturbing the dynamics of periodic orbits with support contained in a small neighborhood and preserving given homoclinic relations. Such results were known in the dissipative setting (though some are quite recent, see Nicolas Gourmelon‘s work). However, the conservative (i.e., volume-preserving or symplectic) versions were often weaker or even lacking. The new preprint fills this gap.
, où 
est une union finie d’obstacles strictement convexes, lisses, sur le bord desquels s’effectue une collision élastique. On suppose également que le temps entre deux collisions successives est borné. La mesure de Liouville (ou plutôt la mesure 
induite sur une hypersurface d’énergie constante, i.e., de vitesse fixée) est invariante pour ce flot 
, en fait ergodique et mélangeante d’après Sinaï. Si 
sont deux fonctions 
sur l’espace des phases, alors 
quand 
. Le problème est d’obtenir une borne de la forme:
et 
.
.
compacte. Soit 
sur un nombre fini de morceaux avec extension. Supposons:
where 
is a compact manifold.
which is (i) invariant: 
; (ii) chain recurrent: for any 
, any 
, there exists a finite sequence 
such that 
; and (iii) whose basin, 
, is a neighborhood of 
. This last property can be stated as: it admits a neighborhood 
such that 
.
-generic diffeomorphism with no topological attractor (i.e., this property holds for all diffeomorphism in a subset containing a dense intersection of countably many open subsets of some fixed open subset of 
).
around a fixed point 
to get:
;
, resp., 
, and close to the unstable, resp. stable, spaces of the linear map.
such that, any center-stable disk of radius bigger than 
and any unstable curve of length at least 
intersect.
such that 
and 
. An application of Zorn’s Lemma shows that all homeomorphisms of compact sets have quasi-attractors.
away from 
. It follows that any quasi-attractor contains 
contracts areas. By Bonatti-Diaz-Pujals, this generically implies that the class is accumulated by sources.
, which is dense by the 
be a group. Given 
and 
, the length 
is the minimum length of a product of elements of 
equal to 
(possibly 
).
.
be the group of 
-smooth diffeomorphisms of a compact manifold 
, then 
and Militon’s proof follows Avila’s.
such that for any sequence of diffeomorphisms 
satisfying 
there exists a finite subset 
for all 
.
denotes the set of smooth diffeomorphisms of 
which are compact supported and are isotopic to the identity through a path of diffeomorphisms with supports all included in a fixed compact set of $\mathbb R^d$.
satisfying 
, 
(and similar conditions on 
), there exists a finite subset ![\ell_S([f_n,g_n])\leq k_n](https://s0.wp.com/latex.php?latex=%5Cell_S%28%5Bf_n%2Cg_n%5D%29%5Cleq+k_n&bg=ffffff&fg=333333&s=0&c=20201002)
for all 
, the set of 
diffeomorphisms of some manifold 
, is a group. It is for instance of interest to know when it is simple.
, the set of all elements that commute with 
that conjugate 
may be very large, e.g., equal to the whole of 
(which is always part of it). Note that such a diffeomorphism cannot be embedded in a smooth flow.
of 
) was one of Smale’s questions for the the twenty-first century.
is open and dense for all 
(
and axiom A with no-cycles, i.e., 
-stable, diffeomorphisms instead of the above ones with strong transversality, i.e., the structurally stable ones.
.
be a pseudo-Anosov on a hyperelliptic translation surface of genus g admitting an involution with 
fixed points). Assume that 
(but approach this value as 
).
where 
is a continuous self-map and 
is a compact metric space. One approach is to “reduce” such systems to symbolic dynamical system, i.e., 
where 
is a closed subset of 
and 
such that 
.
and a continuous surjection 
commuting with the dynamics: 
. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?
.
be a hyperbolic periodic point. Define 
has a transverse point 
. Define 
similarly.
and 
then these two sets are equal (up to a zero volume set) and the restriction of 
for 
. If the exponents along the fiber vanish, then these holonomy preserve the conditional measures. If the base map is hyperbolic and the base measure has a product structure then the conditional measures depend continuously on the base point.
be partially hyperbolic with a central foliation into circles. If 
Jérôme Buzzi's 