Distorted diffeomorphisms and path connectedness of Z^d actions on 1-manifolds
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The document discusses the properties of hyperbolic and non-hyperbolic dynamic systems, focusing on asymptotic distortion and Mather invariants. It includes insights into the behavior of diffeomorphisms, particularly those with periodic points and zero topological entropy, and outlines theorems regarding the construction of diffeomorphisms with specific distortion characteristics. Key results indicate connections between asymptotic distortion, ergodicity, and the behavior of actions under conjugation.
![• 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, ...)](https://image.slidesharecdn.com/bonatti-birthday-210618111751/85/Distorted-diffeomorphisms-and-path-connectedness-of-Z-d-actions-on-1-manifolds-14-320.jpg)
Distorted diffeomorphisms and path connectedness of Z^d actions on 1-manifolds
- 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.