Preliminaries
Probabilistic repellers
Energy estimates
Continuity of Lyapunov exponents
Marcelo Viana
(with C. Bocker, A. Avila and A. Eskin)
deLeónfest, December 2013
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Lyapunov exponents
Consider A1 , . . . , AN ∈ GL(d) and p1 , . . . , pN > 0 with
P
j
pj = 1.
Let (Bn )n be identical independent
random variables in GL(d) with
P
probability distribution µ = j pj δAj . The Lyapunov exponents
1
log kBn · · · B1 k
n
1
λ− (µ) = lim − log k(Bn · · · B1 )−1 k
n
n
λ+ (µ) = lim
n
exist almost surely (Furstenberg, Kesten).
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Continuity theorem
“Theorem”
The functions (Ai ,j , pj )i ,j 7→ λ± are continuous.
d = 2 Carlos Bocker, MV (2010)
d ≥ 2 Artur Avila, Alex Eskin, MV (ongoing)
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Continuity theorem
“Theorem”
The functions (Ai ,j , pj )i ,j 7→ λ± are continuous.
d = 2 Carlos Bocker, MV (2010)
d ≥ 2 Artur Avila, Alex Eskin, MV (ongoing)
The statement extends to the space G(d) of compactly supported
probability measures on GL(d), with a natural topology:
The functions µ 7→ λ± (µ) are continuous on G(d).
The topology is defined by: ν is close to µ if ν is weak∗ -close to µ
and supp ν ⊂ Bǫ (supp µ).
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
“Counterexamples”
The statement sometimes made, that there exist only analytic
functions in nature, is in my opinion absurd. Felix Klein.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
“Counterexamples”
The statement sometimes made, that there exist only analytic
functions in nature, is in my opinion absurd. Felix Klein.
Example
2−1 0
0 −1
For A1 =
A2 =
0 2
1 0
we have λ+ = 0 if p2 > 0 but λ+ = log 2 if p2 = 0.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
“Counterexamples”
The statement sometimes made, that there exist only analytic
functions in nature, is in my opinion absurd. Felix Klein.
Example
2−1 0
0 −1
For A1 =
A2 =
0 2
1 0
we have λ+ = 0 if p2 > 0 but λ+ = log 2 if p2 = 0.
Mañé-Bochi-V:
For any ergodic measure preserving transformation f : M → M,
the continuity points for Lyapunov exponents in the space of 2D
cocycles over f are very special cocycles: either λ± = 0 or the
cocycle is uniformly hyperbolic.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
A partial result
A vector subspace L < Rd is µ-invariant if Aj (L) = L for every j.
Theorem (Furstenberg-Kifer)
P
If µ = j pj δAj is a discontinuity point for λ+ then there exists
β1 < λ+ and a µ-invariant subspace L1 ⊂ Rd such that
limn n1 log k(gn · · · g1 )(v )k ≤ β1 almost surely, for every
v ∈ L1 \ {0}.
limn n1 log k(gn · · · g1 )(v )k = λ+ almost surely, for every
v ∈ Rd \ L1 .
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Examples
Example
3−1 0
2 1
For A1 =
A2 =
p1 = p2 = 1/2.
0 3
0 2−1
Here: L1 = X − axis and β1 = log 2/3 (whereas λ+ = log 3/2).
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Examples
Example
3−1 0
2 1
For A1 =
A2 =
p1 = p2 = 1/2.
0 3
0 2−1
Here: L1 = X − axis and β1 = log 2/3 (whereas λ+ = log 3/2).
Example
3−1 0
2 0
For A1 =
A2 =
0 3
1 2−1
This is a continuity point for λ+ (and λ− ).
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
p1 = p2 = 1/2.
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Examples
Example
3−1 0
2 1
For A1 =
A2 =
p1 = p2 = 1/2.
0 3
0 2−1
Here: L1 = X − axis and β1 = log 2/3 (whereas λ+ = log 3/2).
Example
3−1 0
2 0
For A1 =
A2 =
0 3
1 2−1
This is a continuity point for λ+ (and λ− ).
p1 = p2 = 1/2.
So: discontinuity could only happen if there exists a common
invariant subspace which, in addition, is “mostly contracting”.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Representation of Lyapunov exponents
A probability measure η in PRd is µ-stationary if
X
d
η(E ) =
pj η(A−1
j (E )) for every E ⊂ PR .
j
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Representation of Lyapunov exponents
A probability measure η in PRd is µ-stationary if
X
d
η(E ) =
pj η(A−1
j (E )) for every E ⊂ PR .
j
The stationary measures determine the Lyapunov exponents:
Z
X
pj φ(Aj , v ) dη(v ) : η a µ-stationary measure
λ+ = sup
j
where φ is the expansion function:
φ : GL(d) × PRd → R,
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
φ(g , v ) = log
kg (v )k
.
kv k
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Probabilistic repellers
Example
For A1 =
3−1 0
0 3
A2 =
2 0
0 2−1
p1 = p2 = 1/2
there are two different µ-stationary measures in PR2 , namely, the
Dirac masses at the X -axis and the Y -axis. The one at the X -axis
does not realize the exponent λ+ .
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Probabilistic repellers
Example
For A1 =
3−1 0
0 3
A2 =
2 0
0 2−1
p1 = p2 = 1/2
there are two different µ-stationary measures in PR2 , namely, the
Dirac masses at the X -axis and the Y -axis. The one at the X -axis
does not realize the exponent λ+ .
The invariant subspace L1 = X -axis is a probabilistic repeller. The
ideology of the proof is that such probabilistic repellers should be
negligible for most probability distributions near µ.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
A nonlinear setting
Let M be a compact Riemannian manifold (examples: PRd ,
Grassmannian manifolds) and M be the space of probability
measures on M.
Let G < Diff 1 (M) (e.g. G = GL(d)) and G be the space of
compactly supported probability measures on G .
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
A nonlinear setting
Let M be a compact Riemannian manifold (examples: PRd ,
Grassmannian manifolds) and M be the space of probability
measures on M.
Let G < Diff 1 (M) (e.g. G = GL(d)) and G be the space of
compactly supported probability measures on G .
A point v ∈ M is µ-invariant if g (v ) = v for every g ∈ supp µ.
Then L(µ, v̇ ) = limn n1 log kD(gn · · · g1 )(v )v̇ k exists almost surely
for every non-zero v̇ ∈ Tv M.
We call v µ-expanding if L(µ, v̇ ) > 0 for every v̇ 6= 0.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Instability of µ-expanding points
Theorem
Suppose that v is µ-expanding and (µn )n converges to µ in G. For
each n, let ηn ∈ M be a µn -stationary measure having no atoms in
a fixed neighborhood of v , and assume that (ηn )n converges to
some η ∈ M. Then η({v }) = 0.
This proves continuity of λ+ for all d ≥ 2 when dim L1 = 1. We
are working on a general argument, by induction on the dimension
of the invariant subspace L1 .
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Instability of µ-expanding points
Theorem
Suppose that v is µ-expanding and (µn )n converges to µ in G. For
each n, let ηn ∈ M be a µn -stationary measure having no atoms in
a fixed neighborhood of v , and assume that (ηn )n converges to
some η ∈ M. Then η({v }) = 0.
This proves continuity of λ+ for all d ≥ 2 when dim L1 = 1. We
are working on a general argument, by induction on the dimension
of the invariant subspace L1 .
The original Bocker-V approach in d = 2 is based on a careful
discretization of the phase space PR2 . With Avila and Eskin, we
have been trying with a more direct analysis of the random walk in
continuum space, based on certain energy estimates.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Energy
Given β > 0, the β-energy of a measure ξ on M × M is
Z
Eβ (ξ) = d(x, y )−β dξ(x, y ).
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Energy
Given β > 0, the β-energy of a measure ξ on M × M is
Z
Eβ (ξ) = d(x, y )−β dξ(x, y ).
Let η1 , η2 be measures on M with η1 (M) = η2 (M):
A coupling of (η1 , η2 ) is a measure ξ on M × M that maps to ηj
on the jth coordinate, for j = 1, 2.
Given β > 0, define eβ (η1 , η2 ) = infimum of β-energy Eβ (ξ) over
all couplings ξ. The infimum is attained.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Detecting atoms
Given a measure η on M, define eβ (η) = eβ (η, η). This is finite iff
η has no fat atoms:
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Detecting atoms
Given a measure η on M, define eβ (η) = eβ (η, η). This is finite iff
η has no fat atoms:
η({x}) < 21 η(M) for every x
eβ (η) < ∞
⇒
⇒
eβ (η) < ∞
η({x}) ≤ 12 η(M) for every x
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Detecting atoms
Given a measure η on M, define eβ (η) = eβ (η, η). This is finite iff
η has no fat atoms:
η({x}) < 21 η(M) for every x
eβ (η) < ∞
⇒
⇒
eβ (η) < ∞
η({x}) ≤ 12 η(M) for every x
Lemma
If v is µ-expanding then there exists a neighborhood V of v , a
weak∗ neighborhood V of µ and a constant c > 0 such that
Z
d(g (x), g (y ))−β dν(g ) < (1 − cβ)d(x, y )−β
for every x 6= y in V , every ν ∈ V and every small β > 0.
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents
Preliminaries
Probabilistic repellers
Energy estimates
Manuel de Léon
Feliz Cumpleaños,
Manuel!
Marcelo Viana (with C. Bocker, A. Avila and A. Eskin)
Continuity of Lyapunov exponents