A talk for the conference "Systèmes dynamiques uniformement et non uniformement hyperboliques" honoring Christian Bonatti.

1. Andrés Navas
U. de Santiago (Chile)
Systèmes dynamiques uniformément et non-
uniformément hyperboliques

7. f → var (log (D f )) (Denjoy...)
V ( f ) := lim
𝑛
𝑣𝑎𝑟 (log 𝐷 𝑓 𝑛)
𝑛
 V (f n) = n V(f )
 V (f) is invariant under conjugacy
• V (f ) = infh｛ var(log D(h f h-1))｝

8. For every C2 circle diffeomorphism of irrational rotation number,
the value of V(f ) vanishes (N. 2018).
Remarks:
- This is an analytic consequence of the ergodicity of f (Katok-Herman);
- This implies that such a diffeomorphism approaches the rotation in 1+bv topology by
conjugates (paths of...);
- This result is FALSE for 1+bv diffeomorphisms (Mather non simplicity); the right setting for
the result is the space of diffeomorphisms with absolutely continuous derivative (1+ac).

9. Vanishing of the asymptotic distortion is equivalent to

10. • This case reduces to interval diffeomorphisms up to passing to a finite
power (plus some “details”...).
• If a diffeomorphism f has an hyperbolic periodic point, then V ( f )
is automatically positive.
• Let us consider the space D of interval diffeomorphisms with no fixed
point in the interior and parabolic endpoints (Mather, Yoccoz, ...).

11. V (f ) = var (log DMf )
Here, f belongs to D, and M f denotes its Mather invariant.
• The Mather invariant is a circle diffeomorphism up to pre/post composition
with rotations; it is trivial (i.e. equals the class of rotations) if and only if the
map comes from a C1 vector field;
• The equality above extends to an inequality
| V(f ) - var (log DMf ) | ≤ | log Df (a) | + | log Df (b) |

13. • Given a finitely generated group G, an element g is distorted if its
powers gn can be written as products of o(n) factors (generators).
• Given an arbitraty group, an element g is distorted if it is
distorted inside a finitely generated subgroup.

14. • Question: Given a compact manifold M and r > s, build a Cr diffeomorphism
of M that is Cs distorted yet Cr undistorted.
• Theorem: This exists for r= 2 and M = [a,b]. Moreover, the distortion
achieved is sharp (exponential; Polterovich-Sodin).
Remark. For r=1 this is easier: distorted diffeomorphisms have zero top.
entropy (Calegari-Friedman, Mann, Le Roux, Militon, Rosendal, ...)

15. • Start with a diffeomorphism f that comes from a vector field.
• Perturb it only on a single fundamental domain: F = fg
• The Mather invariant of F becomes nontrivial, hence F has positive asymptotic
distortion, and therefore is C2 undistorted.
• Do this carefully to ensure C1 distortion...
(N. 2019: using a classical
technique of Kopell, Firmo,
Druck, Bonatti, Farinelli...);
(Dinamarca-Escayola 2020:
using direct computations...).
F = f g

16. • The sequence of “good” conjugators for a given f is explicit:

17. • This can be easily transformed into a path of conjugators.
• Moreover, similar formulae work for Zd actions:
• This naturally leads to path-connectedness for Zd actions provided
group elements have zero asymptotic distortion.

18. The space of Zd actions by C1+ac diffeomorphisms of a 1-manifold is path connected.
Comments:
• Related to an old question by Rosenberg; results by Bonatti-Eynard (connec-tedness,
interval, 𝐶 ∞) and N. (path connectedness, 𝐶 1 ).
 The difficult case is that of the interval with elements having positive asymptotic distortion
though parabolic fixed points.
 There may be infinitely many fixed points; this requires deforming the restriction of the
action to each subinterval with no fixed points in a controlled way.

19. • If the asymptotic distortion is positive then the Mather invariant is nontrivial, and the centralizer is cyclic
(Mather, Yoccoz...).
• Zd = < f1 , f2 , ... , fd >; there is F such that 𝑓𝑖 = 𝐹 𝑛𝑖
• The naive idea consists in just deforming F (e.g. by linear isotopy) and the action accordingly.
Warning !
It may happen that F is far from the identity
despite each fi is close to it (Sergeraert, Eynard).

20. Put the action in right coordinates before deforming it:
• Perform the conjugacy trick via the conjugating maps hn
(this is a well-behaved process).
• After this, perform the linear isotopy deformation of the generator h
Fh-1 and deform the action accordingly.

21. AND MANY THANKS...