Slides of a talk for the Virtual thematic workshop in Math. Sciences "Dynamical Systems": https://www.youtube.com/watch?v=ukyCxEUZQm8&feature=youtu.be

1. ON
GROUPS
AND
DYNAMICS

2. “Les mathématiques ne sont qu'une histoire de
groupes” (Henri Poincaré)
Cayley: A group is a set endowed with
a multiplication together with certain rules.
Cayley: “Every group acts”.
Cayley graph:
vertives: group elements
edges: between elements
differing by a generator

3. “Les mathématiques ne sont qu'une histoire de
groupes” (Henri Poincaré)
Cayley: A group is a set endowed with
a multiplication together with certain rules.
Cayley: “Every group acts”.
Cayley graph:
vertives: group elements
edges: between elements
differing by a generator

4. If a group acts nicely on a nice space, then this should reveal
something about its algebraic structure.
Example (exercise): If a group acts freely by circle homeomorphisms,
then it is Abelian (Hölder).
Warning: Every countable group arises as a subgroup of the group of
homeomorphisms of the Cantor set:

5. Burnside: If a finitely generated group is such that
every element has finite order, is the group finite ?
• B (n) = < a, b : wn = id >
• B (2), B (3), B (4) and B (6) are finite
• B (7) should be infinite (Gromov)
• B (5) should be infinite (Zelmanov)
• B (n) is inifinite for odd n > 666
Question: Is every Burnside group of
homeomorphisms of the sphere finite?
(Hurtado, Kocsard, Rodríguez-Hertz;
Guelman, Liousse; Conejeros).

6. Understanding group-theoretical properties of
diffeomorphisms gives relevant dynamical
information ont the map
• Nancy Kopell:
Commuting Diffeomorphisms
(1968).
Smale,
Mather,
Palis-Yoccoz,
Bonatti-Crovisier-Wilkinson.

7. Kopell's lemma
• Theorem (N, 2008): There is no group of intermediate growth of
C1+e diffeomorphisms of neither the circle nor the interval (and
this is false in class C1).
• Theorem (Kim-Koberda; Mann-Wolf): for every r > s there exists a
finitely generated group of Cs diffeomorphisms of the interval that
does not embed into the group of Cr diffeomorphisms.

8. Distorted diffeomorphisms
• An element f of a finitely generated group is distorted if the world-length of fn
grows sublinearly on n.
• An element f of a general group G is distorted if it is distorted inside a finitely
generated subgroup of G.
Example: g f g-1 = f2 implies gn f g-n = 𝒇 𝟐 𝒏
( f : x → x+1 ; g : x → 2x )
Example: If a diffeomorphism has an hyperbolic fixed points
(positive Lyapunov exponents, positive topological entropy),
then it is undistorted inside the group of C1 diffeomorphisms.

9. A final question on distortion
• Given r > s and a compact manifold M, does there exist a Cr
diffeomorphism of M that is undistorted in G = Diffr(M) but distorted
in G = Diffs(M) ?
• Theorem (N; Dinamarca, Escayola): YES for M the closed interval, r =
2 and s =1 (s = 1 + e).
• We even don't know what happens for the case of the circle (Avila,
Mather, N)...

10. An idea for the proof
(a Lyapunov exponent for higher derivatives?)
• Consider the variation of the logarithm of the derivative:
Consider the asymptotic variation: V ( f ) := lim
𝒗𝒂𝒓 ( 𝒍𝒐 𝒈 𝑫𝒇 𝒏 )
𝒏
If then f is undistorted in the group of C2
diffeomorphisms (work with Hélène Eynard-Bontemps).