Lyapunov%20Exponents

1
Lyapunov Exponents
Linear cocycle over ?
continuous and linear (isomorphism) on fibers
2
Lyapunov Exponents
Linear cocycle over ?
continuous and linear (isomorphism) on fibers
A M G , G GL(d) Fn(x,v) (?n(x) ,
An(x)v)
?
where
An(x) A(?n-1x) ... A(?x) A(x)
with
3
Oseledets thm
for any ƒ-invariant probability ? and ?-almost
every x?M
k k(x)
? ?1(x) gt ... gt ?k(x)
4
Oseledets thm
for any ƒ-invariant probability ? and ?-almost
every x?M
k k(x)
? ?1(x) gt ... gt ?k(x)
for all v ? Ex\0
i
are just measurable
5

1. How do the ?i depend on F ?
2. How often do ?i 0 ?

Cr (M,G) r ? 0
6

1. How do the ?i depend on F ?
2. How often do ?i 0 ?

Cr (M,G) r ? 0
Ex (dynamical cocycles)
E TM
F D?
r ? 1
7
Ex (random matrices)
Let ?0 , ?1 , ... be i. i. d. random
variables in SL(d), with probability distribution
?. What can be said of
8
Ex (random matrices)
Let ?0 , ?1 , ... be i. i. d. random
variables in SL(d), with probability distribution
?. What can be said of
Furstenberg if supp(?) is rich enough
then gt 0 almost surely
9
Ex (random matrices)
Let ?0 , ?1 , ... be i. i. d. random
variables in SL(d), with probability distribution
?. What can be said of
lim
log ?n-1 ... ?1 ?0 ?
Furstenberg if supp(?) is rich enough
then gt 0 almost surely
then ?1
for the corresponding cocycle
10
Ex (random matrices)
Let ?0 , ?1 , ... be i. i. d. random
variables in SL(d), with probability distribution
?. What can be said of
An (x) , x (?n)n
Furstenberg if supp(?) is rich enough
then gt 0 almost surely
then ?1
for the corresponding cocycle
11
Ex Let M S1
?(0) 0
? ergodic with supp ? M
A M SL(2, )
A0 hyperbolic
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Ex Let M S1
?(0) 0
? ergodic with supp ? M
A M SL(2, )
A0 hyperbolic
Thm Assume deg(?) ? 2,3. There exists a C0
neigbourhood U of A such that

1. for generic (dense G?) B ?U ?1
   0 ( k 1) a.e.
2. for every Hölder continuous B ?U
   ?1 gt 0 gt ?2 - ?1 a.e.

?
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Thm1 (Bochi, V)
d 2 Mañé, Bochi
Assume (?,?) is ergodic, and G ? SL (d)
acts transitively on the projective space.
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Thm1 (Bochi, V)
d 2 Mañé, Bochi
Assume (?,?) is ergodic, and G ? SL (d)
acts transitively on the projective space. Then
there exists a residual R ? C0 (M, G) s.t. for
every A ? R either a)
all Lyapunov exponents are equal to zero a.e.
15
Thm1 (Bochi, V)
d 2 Mañé, Bochi
Assume (?,?) is ergodic, and G ? SL (d)
acts transitively on the projective space. Then
there exists a residual R ? C0 (M, G) s.t. for
every A ? R either a) or b)
all Lyapunov exponents are equal to zero
a.e. the Oseledets splitting is dominated
16
Thm1 (Bochi, V)
d 2 Mañé, Bochi
Assume (?,?) is ergodic, and G ? SL (d)
acts transitively on the projective space. Then
there exists a residual R ? C0 (M, G) s.t. for
every A ? R either a) or b)
- -
all Lyapunov exponents are equal to zero
a.e. the Oseledets splitting is dominated
it admits a continuous extension to
supp(?) angles are bounded from zero
17
Ex M S1
? ergodic with supp (?) M
A M SL(2, )
such that deg(?) 1 2 deg (A) Then,
generically in the homotopy class of A, all
Lyapunov exponents are zero a.e.
18
Ex M S1
? ergodic with supp (?) M
A M SL(2, )
such that deg(?) 1 2 deg (A) Then,
generically in the homotopy class of A, all
Lyapunov exponents are zero a.e.
Reason the cocycle has no
continuous invariant sub-bundle
?
19
Comments on the proof of thm1
?1(x) gt ... gt ?k(x)
Lyapunov exps
?
?
?1(x) gt ... gt ?k(x)
Lyapunov exps
counted with multiplicity dim
?i(A) ?M ?i(x) d ?
?
?
20
Comments on the proof of thm1
?1(x) gt ... gt ?k(x)
Lyapunov exps
?
?1(x) gt ... gt ?k(x)
?
Lyapunov exps
counted with multiplicity dim
?i(A) ?M ?i(x) d ?
?
?
Thm2 (Bochi, V)
A is a point of continuity of C0(M,G) B
(?1(B), ... , ?d(B))
?
?
?
?
a) all Lyapunov exponents are equal to zero
a.e. or else b) the Oseledets decomposition is
dominated
21
Comments on the proof of thm1
?1(x) gt ... gt ?k(x)
Lyapunov exps
?
?1(x) gt ... gt ?k(x)
?
Lyapunov exps
counted with multiplicity dim
?i(A) ?M ?i(x) d ?
?
?
Thm2 (Bochi, V)
A is a point of continuity of C0(M,G) B
(?1(B), ... , ?d(B))
?
?
?
?
a) all Lyapunov exponents are equal to zero
a.e. or else b) the Oseledets decomposition is
dominated
continuity points contains dense G?
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Ex M S1
? M M , ? (?) ? ? , ? ?
Schrödinger cocycle
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Ex M S1
? M M , ? (?) ? ? , ? ?
Schrödinger cocycle
A is a point of continuity of Lyapunov
exponents
?
either the exponents are zero or E ? spectrum
of associated Schrödinger operator
24
What about dynamical cocycles ?
F D? M manifold
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What about dynamical cocycles ?
F D? M manifold
Thm3 (Bochi, V)
d 2 Mañé, Bochi
1
There exists a residual R ? Diff? (M) in the
space of volume preserving diffeomorphisms,
such that for every ??R and ?- almost every x?M
either a) all Lyapunov exponents are
zero at x or b) the Oseledets
decomposition is dominated on the
orbit of x
26
Thm4 (Bochi, V)
1
There is a residual R ? Symp? (M) in the space
of symplectic diffeomorphisms such that for every
??R either a) almost every point has zero
as Lyapunov exponent (multiplicity ?
2) or b) ? is Anosov very strong
restrictions on the manifold !
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What about A ? Cr (M,G)
r1
? ? Diff? (M) F D?
for r gt 0 ?
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What about A ? Cr (M,G)
r1
? ? Diff? (M) F D?
for r gt 0 ?

• Assume (?, ? ) is hyperbolic
• (non-uniformly)
• all exponents of DF non-zero
• ? ergodic, non-atomic, with local
  product structure
• G SL(d) or Symp (2d)

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Thm5
For every r gt 0 the set of A ? Cr (M,G) with
positive Lyapunov exponents contains an open
dense set. Moreover, its complement has ? -
codimension.
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Thm5
For every r gt 0 the set of A ? Cr (M,G) with
positive Lyapunov exponents contains an open
dense set. Moreover, its complement has ? -
codimension.
it is contained in finite unions of closed
submanifolds with arbitrary codimension

• Bonatti, Gomez-Mont, V
• same conclusion when
• ? ? Axiom A and the cocycle
• is partially hyperbolic

31
One key ingredient
? M M
Consider uniformly
expanding and supp (?) M and r Lipschitz
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One key ingredient
? M M
Consider uniformly
expanding and supp (?) M and r Lipschitz
The cocycle A is bundle-free if ?? ? 1 there
exists no Lipschitz map M ? x ?1(x) , ...
, ??(x) ?(x)
distinct points in pd-1
invariant under A A(x) . ? (x) ?(?(x)) ?
x ? M
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Ex M S1
such that deg (?) 1 ? 2 deg (A) Then A is
bundle-free
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Ex M S1
such that deg (?) 1 ? 2 deg (A) Then A is
bundle-free
Thm
Assume A ? CLipschitz satisfies

• A is bundle-free
• there is p ? Fix(?k), k ? 1 such that all
  eigenvalues of Ak(p) have distinct norms
• Then ?1(A) gt 0.

Both conditions contain open and dense set,
the complement has ?-codimension
35
What are the continuity points of Lyapunov
exponents in Cr (M,G) ?
36
What are the continuity points of Lyapunov
exponents in Cr (M,G) ?
ongoing , Avila, Bochi, V

• bundle-free cocycles are continuity
  points
• discontinuities do exist, at least
  for small r gt 0

37
What are the continuity points of Lyapunov
exponents in Cr (M,G) ?
ongoing , Avila, Bochi, V

• bundle-free cocycles are continuity
  points
• discontinuities do exist, at least
  for small r gt 0

Lower estimates of exponents ?