Optimization of Lyapunov Exponents

RDS and MET Workshop at BIRS, Banff, Canada

Jairo Bochi (Catholic Univ. of Chile, Santiago)

January 22, 2015

Jairo Bochi Optimization of Lyapunov Exponents 1 / 48

Joint work with

Michał Rams

(Polish Academy of Sciences, Warsaw)

Plan of the talk

1 Thanks to the organizers.2 “Classical” ergodic optimization3 Non-commutative ergodic optimization: what we would

like to do4 Setting: One-step cocycles and domination5 A useful tool: Barabanov functions6 The main result7 Strategy of the proof

ERGODIC OPTIMIZATION:

FROM BIRKHOFF AVERAGES TO LYAPUNOV EXPONENTS

Ref.: Survey by O. Jenkinson (DCDS, 2006)

“Classical” ergodic optimizationFix a continuous dynamics T : X→ X on a compact space X

Let M(T) be set of T-invariant Borel probabilities.

Given a continuous function f : X→ R, let

M(f ) := supμ∈M(T)

f dμ.

An invariant measure for which the sup is attained is calledmaximizing for f . (They always exist.)

“Classical” Ergodic Optimization is concerned aboutdetermining M(f ), describing the maximizing measures μ,their properties etc.

Rem.: Connections with Classical Mechanics(Aubry–Mather. . . ), Thermodynamical Formalism (zerotemperature limits), Multifractal Analysis . . .

“Classical” ergodic optimizationFix a continuous dynamics T : X→ X on a compact space X

Let M(T) be set of T-invariant Borel probabilities.

Given a continuous function f : X→ R, let

M(f ) := supμ∈M(T)

f dμ.

An invariant measure for which the sup is attained is calledmaximizing for f . (They always exist.)

“Classical” Ergodic Optimization is concerned aboutdetermining M(f ), describing the maximizing measures μ,their properties etc.

Rem.: Connections with Classical Mechanics(Aubry–Mather. . . ), Thermodynamical Formalism (zerotemperature limits), Multifractal Analysis . . .

The place where maximizing measures live

Theorem (Many authors(*), 90’s, 00’s)Suppose T : X→ X is “hyperbolic” and f is Hölder continuous.Then there exists a compact T-invariant set K ⊂ X (called a“Mather set”) such that:

μ ∈M(T) is maximizing ⇔ suppμ ⊂ K .

(*) Coelho, Conze – Guivarc’h, Savchenko, Mañé, Fathi, Contreras – Lopes –Thieullen, Bousch.

N.B.: The result is false for C0 functions; in this case there areexamples where the maximizing measure is unique and hasfull support.

The place where maximizing measures live

Theorem (Many authors(*), 90’s, 00’s)Suppose T : X→ X is “hyperbolic” and f is Hölder continuous.Then there exists a compact T-invariant set K ⊂ X (called a“Mather set”) such that:

μ ∈M(T) is maximizing ⇔ suppμ ⊂ K .

(*) Coelho, Conze – Guivarc’h, Savchenko, Mañé, Fathi, Contreras – Lopes –Thieullen, Bousch.

N.B.: The result is false for C0 functions; in this case there areexamples where the maximizing measure is unique and hasfull support.

Proof of the existence of Mather setsSuppose f̃ is cohomologous to f , i.e.,

f̃ = f + h ◦ T − h for some h

ThenM(f ) = sup

μ∈M(T)

f dμ = supμ∈M(T)

f̃ dμ = M(f̃ ).

μ is maximizing for f ⇐⇒ μ is maximizing for f̃ .

Theorem (Mañé(*) Lemma or Revelation Lemma)Suppose T : X→ X is “hyperbolic” and f is Hölder continuous.Then ∃ f̃ cohomologous to f such that f̃ ≤M(f̃ ).

Then the maximizing measures are exactly those who live inthe (Mather set) K := {x ∈ X; f̃ (x) = M(f̃ )}.

Proof of the existence of Mather setsSuppose f̃ is cohomologous to f , i.e.,

f̃ = f + h ◦ T − h for some h

ThenM(f ) = sup

μ∈M(T)

f dμ = supμ∈M(T)

f̃ dμ = M(f̃ ).

μ is maximizing for f ⇐⇒ μ is maximizing for f̃ .

Theorem (Mañé(*) Lemma or Revelation Lemma)Suppose T : X→ X is “hyperbolic” and f is Hölder continuous.Then ∃ f̃ cohomologous to f such that f̃ ≤M(f̃ ).

Then the maximizing measures are exactly those who live inthe (Mather set) K := {x ∈ X; f̃ (x) = M(f̃ )}.

TheoremIf T is “expanding” then for generic Lipschitz functions f :

(several authors) There is a unique maximizingmeasure.

(Morris 2008) This measure actually has zero entropy.

(Contreras, preprint 2013) This measure is actuallysupported on a periodic orbit.

Related results on periodicity of maximizing measures by:

Hunt–Ott’96 (experimental), Yuan–Hunt’99, Contreras–Lopes-Thieullen’01,Bousch’01, Quas–Siefken’12. The latter deals with spaces ofsuper-continuous functions over the shift.

Theorem (B.–Zhang, preprint ArXiv 2015)On spaces of sufficiently super-continuous functions, theconclusion of Quas–Siefken–Contreras is not only topologicallygeneric but prevalent (has “full probability”).

Non-commutative Ergodic Optimization

What if instead of optimizing integrals (Birkhoff averages), weoptimize Lyapunov exponents instead?

Lyapunov exponents (2× 2 case)Let T : X→ X be a homeomorphism of a compact space. LetA : X→ GL(2,R) be continuous. The pair (T,A) is called acocycle. Consider “Birkhoff-like” products:

A(n)(x) := A(Tn−1x) · · ·A(Tx)A(x).

The upper and lower Lyapunov exponents at the point x (ifthey exist) are:

λ1(x) := limn→+∞

A(n)(x)

λ2(x) := limn→+∞

[A(n)(x)]−1

A(n)(x)' enλ1(x)

enλ2(x) '

Lyapunov exponents (2× 2 case)Let T : X→ X be a homeomorphism of a compact space. LetA : X→ GL(2,R) be continuous. The pair (T,A) is called acocycle. Consider “Birkhoff-like” products:

A(n)(x) := A(Tn−1x) · · ·A(Tx)A(x).

The upper and lower Lyapunov exponents at the point x (ifthey exist) are:

λ1(x) := limn→+∞

A(n)(x)

λ2(x) := limn→+∞

[A(n)(x)]−1

A(n)(x)' enλ1(x)

enλ2(x) '

Oseledets Theorem

Theorem (Invertible 2× 2 Oseledets)Let μ be an ergodic probability measure for T. Then thereexist λ1(μ) ≥ λ2(μ) such that λi(x) = λi(μ) for μ-a.e. x.

Moreover, if λ1(μ) > λ2(μ) then there exists an “Oseledetssplitting” R2 = E1(x)⊕ E2(x), defined for μ-a.e. x and such thatfor each i = 1,2:

the spaces Ei vary measurably w.r.t. x;

the spaces Ei are equivariant: A(x)(Ei(x)) = Ei(Tx);

limn→±∞

nlog ‖A(n)(x)v‖ = λi(μ), ∀v ∈ Ei(x)r {0} (i = 1,2).

Oseledets Theorem

Theorem (Invertible 2× 2 Oseledets)Let μ be an ergodic probability measure for T. Then thereexist λ1(μ) ≥ λ2(μ) such that λi(x) = λi(μ) for μ-a.e. x.

Moreover, if λ1(μ) > λ2(μ) then there exists an “Oseledetssplitting” R2 = E1(x)⊕ E2(x), defined for μ-a.e. x and such thatfor each i = 1,2:

the spaces Ei vary measurably w.r.t. x;

the spaces Ei are equivariant: A(x)(Ei(x)) = Ei(Tx);

limn→±∞

nlog ‖A(n)(x)v‖ = λi(μ), ∀v ∈ Ei(x)r {0} (i = 1,2).

Optimization of Lyapunov exponents

Fixed cocycle (T,A) as above, let:

λ⊤1 := supμ∈Merg(T)

λ1(μ) , λ⊥1 := infμ∈Merg(T)

λ1(μ) .

(Rem.: The λ2 case can be reduced to the λ1 case by inverting time andsign.)

Questions:

Are the sup and inf above attained? That is, are thereμ ∈Merg(T) such that λ1(μ) = λ⊤1 or λ⊥1?How are those measures? What are their entropies,dimensions, etc?Are they unique?Are they characterized by their support?

λ1(μ) .

Questions:

λ1(μ) .

Questions:

Are the sup and inf above attained? That is, are thereμ ∈Merg(T) such that λ1(μ) = λ⊤1 or λ⊥1?

How are those measures? What are their entropies,dimensions, etc?Are they unique?Are they characterized by their support?

λ1(μ) .

Questions:

Are the sup and inf above attained? That is, are thereμ ∈Merg(T) such that λ1(μ) = λ⊤1 or λ⊥1?How are those measures? What are their entropies,dimensions, etc?

Are they unique?Are they characterized by their support?

λ1(μ) .

Questions:

Are the sup and inf above attained? That is, are thereμ ∈Merg(T) such that λ1(μ) = λ⊤1 or λ⊥1?How are those measures? What are their entropies,dimensions, etc?Are they unique?

Are they characterized by their support?

λ1(μ) .

Questions:

Summary of the main result (B., Rams)

We will exhibit a natural open class of 2× 2 cocycles for whichthe Lyapunov-maximizing and -minimizing measures havezero entropy.

An important source of ideas:

paper by T. Bousch, J. Mairesse (JAMS, 2002).

A SPECIAL SITUATION:

ONE-STEP COCYCLES

One-step cocycles

A cocycle (T,A) is one-step if:

T : X→ X is the two-sided full shift on k ≥ 2 symbols, so

(ξi)i∈Z; ξi ∈ {1,2, . . . ,k}o

A : X→ GL(2,R) only depends on the zeroth symbol. Inother words, there is a list of matrices A1, . . . , Ak such that

ξ = (ξi)i∈Z ⇒ A(ξ) = Aξ0

(In particular, A is locally constant.)

Equivalent setting: Linear IFS’s (iterated function systems).

One-step cocycles

(ξi)i∈Z; ξi ∈ {1,2, . . . ,k}o

ξ = (ξi)i∈Z ⇒ A(ξ) = Aξ0

One-step cocycles

(ξi)i∈Z; ξi ∈ {1,2, . . . ,k}o

ξ = (ξi)i∈Z ⇒ A(ξ) = Aξ0

Joint spectral radius and subradius

Equivalent elementary definitions of λ⊤1, λ⊥1 for one-stepcocycles:

λ⊤1 = limn→∞

nlog sup

i1,...,in

‖Ai1 . . .Ain‖ ,

λ⊥1 = limn→∞

nlog inf

i1,...,in‖Ai1 . . .Ain‖ .

For one-step cocycles, the numbers eλ⊤1 and eλ

⊥1 are known as

joint spectral radius (Rota, Strang 1960) and joint spectralsubradius (Gurvits, 1995) of the set of matrices {A1, . . . ,Ak}.

Applications in wavelets, control theory . . .

Joint spectral radius and subradius

Equivalent elementary definitions of λ⊤1, λ⊥1 for one-stepcocycles:

λ⊤1 = limn→∞

nlog sup

i1,...,in

‖Ai1 . . .Ain‖ ,

λ⊥1 = limn→∞

nlog inf

i1,...,in‖Ai1 . . .Ain‖ .

For one-step cocycles, the numbers eλ⊤1 and eλ

⊥1 are known as

joint spectral radius (Rota, Strang 1960) and joint spectralsubradius (Gurvits, 1995) of the set of matrices {A1, . . . ,Ak}.

Applications in wavelets, control theory . . .

The finiteness conjecture on the Joint spectralradius

Finiteness conjecture (1995): For one-step cocycles, therealways exists a Lyapunov-maximizing measure supported ona periodic orbit.

The conjecture was shown to be false by Bousch andMairesse (2002); their counterexamples are pairs of matricesin GL(2,R) such that the maximizing measures are sturmianbut not periodic.

The finiteness conjecture on the Joint spectralradius

Finiteness conjecture (1995): For one-step cocycles, therealways exists a Lyapunov-maximizing measure supported ona periodic orbit.

The conjecture was shown to be false by Bousch andMairesse (2002); their counterexamples are pairs of matricesin GL(2,R) such that the maximizing measures are sturmianbut not periodic.

Domination for one-step cocycles

An one-step cocycle is called dominated if the Lyapunovexponents are uniformly separated, for all invariant measures.In other words,

(λ1 − λ2)⊥(A) := inf

μ∈Merg(T)(λ1(A, μ)− λ2(A, μ))

is positive.

!4This definition applies only to one-step cocycles.

Example (Perron–Frobenius theorem)Each Ai is an (entrywise) positive matrix ⇒ the cocycle isdominated.

Domination for one-step cocycles

An one-step cocycle is called dominated if the Lyapunovexponents are uniformly separated, for all invariant measures.In other words,

(λ1 − λ2)⊥(A) := inf

μ∈Merg(T)(λ1(A, μ)− λ2(A, μ))

is positive.

!4This definition applies only to one-step cocycles.

Example (Perron–Frobenius theorem)Each Ai is an (entrywise) positive matrix ⇒ the cocycle isdominated.

Domination for one-step cocycles: notes

Recall: An one-step cocycle is called dominated if(λ1 − λ2)

⊥ > 0.

For one-step cocycles (only!), the definition above isequivalent to the more usual condition that the oneOseledets direction “dominates” the other.

In the area-preserving case (2× 2 matrices withdeterminant ±1), domination is equivalent to the familiarnotion of uniform hyperbolicity.Domination is a natural weakening of uniformhyperbolicity and is important smooth dynamics (e.g. inpartial hyperbolicity).It’s also weaker than exponential dichotomy.

⊥ > 0.

For one-step cocycles (only!), the definition above isequivalent to the more usual condition that the oneOseledets direction “dominates” the other.In the area-preserving case (2× 2 matrices withdeterminant ±1), domination is equivalent to the familiarnotion of uniform hyperbolicity.

Domination is a natural weakening of uniformhyperbolicity and is important smooth dynamics (e.g. inpartial hyperbolicity).It’s also weaker than exponential dichotomy.

⊥ > 0.

For one-step cocycles (only!), the definition above isequivalent to the more usual condition that the oneOseledets direction “dominates” the other.In the area-preserving case (2× 2 matrices withdeterminant ±1), domination is equivalent to the familiarnotion of uniform hyperbolicity.Domination is a natural weakening of uniformhyperbolicity and is important smooth dynamics (e.g. inpartial hyperbolicity).It’s also weaker than exponential dichotomy.

Domination for one-step cocycles: geometricalviewpointLet P1 be the projective space (lines through the origin in R2).

A multicone for {A1, . . . ,Ak} is an open set M P1 such that

M has a finite number of connected components, and theyhave disjoint closures.∀ i, the closure of the image Ai(M) is contained in M.

ExampleEach Ai is an (entrywise) positive matrix ⇒ the first quadrantis a multicone.

Theorem (Avila, B., Yoccoz)An one-step cocycle is dominated IFF it has a multicone.

Consequence: domination is an open condition. (This is alsotrue for more general cocycles.)

M has a finite number of connected components, and theyhave disjoint closures.

∀ i, the closure of the image Ai(M) is contained in M.

The simplest example where the multicone cannot be takenas a cone is:

A1 = A, A2 = B.

Examples with complicated combinatorics: see [Avila–B.–Yoccoz]

BARABANOV FUNCTIONS

FOR DOMINATED ONE-STEP COCYCLES:

An analogue of the Mañé revelation lemma

Barabanov functionsAssume given an one-step dominated cocycle, with amulticone M. Let

−→M := {v ∈ R2; v = 0 or R · v ∈M} be the

“support” of M.

Proposition (Existence of a Barabanov function)

There exists a continuous function |||·||| :−→M → [0,∞) such that

for every v ∈−→M, t ∈ R we have

|||tv||| = |t| |||v||| (homogeneity);

maxi∈{1,...,k}

|||Aiv||| = eλ⊤1 |||v||| (main property).

We call |||·||| an upper Barabanov function.

Consequence: For any v ∈−→M , we can find recursively

symbols i1, i2, . . . ∈ {1, . . . ,k} such that

|||Ain · · ·Ai1(v)||| = enλ⊤1 |||v|||.

−→M := {v ∈ R2; v = 0 or R · v ∈M} be the

“support” of M.

maxi∈{1,...,k}

|||Ain · · ·Ai1(v)||| = enλ⊤1 |||v|||.

−→M := {v ∈ R2; v = 0 or R · v ∈M} be the

“support” of M.

maxi∈{1,...,k}

|||Ain · · ·Ai1(v)||| = enλ⊤1 |||v|||.

−→M := {v ∈ R2; v = 0 or R · v ∈M} be the

“support” of M.

maxi∈{1,...,k}

|||Ain · · ·Ai1(v)||| = enλ⊤1 |||v|||.Jairo Bochi Optimization of Lyapunov Exponents 23 / 48

Barabanov functions

Upper Barabanov function:

|||tv||| = |t| |||v|||, maxi∈{1,...,k}

|||Aiv||| = eλ⊤1 |||v|||.

Actually (even without domination) there is a true norm|||·||| in R2 with the properties above; it is called Barabanovnorm (1988).

There is also a lower Barabanov function |||·|||′ such that

|||tv|||′ = |t| |||v|||′, mini∈{1,...,k}

|||Aiv|||′ = eλ⊥1 |||v|||′ .

In this case the domination hypothesis is really needed.Higher dimension: see [B.–Morris, Proc. LMS (to appear)].

Barabanov functions

Upper Barabanov function:

|||tv||| = |t| |||v|||, maxi∈{1,...,k}

|||Aiv||| = eλ⊤1 |||v|||.

Actually (even without domination) there is a true norm|||·||| in R2 with the properties above; it is called Barabanovnorm (1988).There is also a lower Barabanov function |||·|||′ such that

|||tv|||′ = |t| |||v|||′, mini∈{1,...,k}

|||Aiv|||′ = eλ⊥1 |||v|||′ .

In this case the domination hypothesis is really needed.Higher dimension: see [B.–Morris, Proc. LMS (to appear)].

Proof of existence of Barabanov functions

We apply Schauder fixed point theorem to an appropriatespace of Lispchitz functions (with respect to the Hilbertmetric) on the multicone . . .

BTW, this proof also gives a Lipschitz property for |||·|||, which isalso useful.

Mather setsPropositionIf an one-step cocycle is dominated then there are nonemptycompact shift-invariant sets K⊤, K⊥ ⊂ kZ s.t. for everyshift-invariant measure μ,

suppμ ⊂ K⊤ ⇔ λ1(μ) = λ⊤1, suppμ ⊂ K⊥ ⇔ λ1(μ) = λ⊥1

(In particular, there exist Lyapunov-optimizing measures.)

Proof:

K⊤ :=n

ξ ∈ X; u ∈ E1(ξ), n ∈ Z ⇒ |||A(n)(ξ)(u)||| = enλ⊤1 |||u|||o

Rem.: The proposition above should also follow from the “classical”(commutative) theory applied to the function f := log ‖A|E1‖. Anyway, ourBarabanov functions with be fundamental for what we will do next.

Mather setsPropositionIf an one-step cocycle is dominated then there are nonemptycompact shift-invariant sets K⊤, K⊥ ⊂ kZ s.t. for everyshift-invariant measure μ,

suppμ ⊂ K⊤ ⇔ λ1(μ) = λ⊤1, suppμ ⊂ K⊥ ⇔ λ1(μ) = λ⊥1

(In particular, there exist Lyapunov-optimizing measures.)

Proof:

K⊤ :=n

ξ ∈ X; u ∈ E1(ξ), n ∈ Z ⇒ |||A(n)(ξ)(u)||| = enλ⊤1 |||u|||o

Rem.: The proposition above should also follow from the “classical”(commutative) theory applied to the function f := log ‖A|E1‖. Anyway, ourBarabanov functions with be fundamental for what we will do next.

OUR MAIN RESULT

Non-overlapping conditionWe say that the matrices A1, . . . ,Ak satisfy forwardnonoverlapping condition (NOC+) if they admit a multicone Msuch that

i 6= j ⇒ Ai(M) ∩ Aj(M) =∅.

We say that the matrices A1, . . . ,Ak satisfy backwardsnonoverlapping condition (NOC−) if {A−1

1 , . . . ,A−1k } satisfy the

forward nonoverlapping condition.

If both conditions hold then we say that the matrices satisfythe nonoverlapping condition (NOC).

Example: The one in a previous figure:

i 6= j ⇒ Ai(M) ∩ Aj(M) =∅.

The main theorem

Theorem (B., Rams)If an one-step cocycle is dominated and satisfies thenon-overlapping condition

then the sets K⊤, K⊥ have zerotopological entropy.

In particular, the optimizing measures have zero entropy.

The main theorem

Theorem (B., Rams)If an one-step cocycle is dominated and satisfies thenon-overlapping condition then the sets K⊤, K⊥ have zerotopological entropy.

In particular, the optimizing measures have zero entropy.

Remarks

The zero entropy measures that appear in the theoremare not necessarily supported on periodic orbits(∃ examples by Bousch–Mairesse and others).Neither they are necessarily unique.The non-overlapping condition is indeed required: forexample, if A1 = A2 then there are optimal measures withpositive entropy.We think that a more general theorem holds, with T =subshift of finite type, A = locally constant cocycle, but wehaven’t checked the details.

QuestionsCan we replace the non-overlapping condition by aweaker hypothesis (preferably “typical” amongdominated cocycles)?

Is there a result of this kind for cocycles that are notlocally constant?(We believe we know what should replace Barabanov functions andhow to construct them.)

What about the non-dominated case? (Maybe therestriction of the cocycle to K⊤ is typically dominated. . . )“Generic finiteness conjecture”: Is is typical forLyapunov-maximizing measures to be periodic?In other words, does a matrix version of Contrerastheorem hold?What about higher dimension?

There is much more to be done in this subject!

QuestionsCan we replace the non-overlapping condition by aweaker hypothesis (preferably “typical” amongdominated cocycles)?Is there a result of this kind for cocycles that are notlocally constant?(We believe we know what should replace Barabanov functions andhow to construct them.)

What about the non-dominated case? (Maybe therestriction of the cocycle to K⊤ is typically dominated. . . )

“Generic finiteness conjecture”: Is is typical forLyapunov-maximizing measures to be periodic?In other words, does a matrix version of Contrerastheorem hold?What about higher dimension?

What about the non-dominated case? (Maybe therestriction of the cocycle to K⊤ is typically dominated. . . )“Generic finiteness conjecture”: Is is typical forLyapunov-maximizing measures to be periodic?In other words, does a matrix version of Contrerastheorem hold?

What about higher dimension?

There is much more to be done in this subject!Jairo Bochi Optimization of Lyapunov Exponents 31 / 48

PROOF OF THE THEOREM

Oseledets directions

Given a bi-infinite word ξ ∈ X = kZ, split it asξ = (ξ− , ξ+) ∈ kZ− × kZ+, where Z− = {. . . ,−2,−1} andZ+ = {0,1, . . .}.

The first Oseledets direction only depends on the past, whilethe second one only depends on the future:

E1(ξ) = E1(ξ−), E2(ξ) = E2(ξ+).

Oseledets directions

Given a bi-infinite word ξ ∈ X = kZ, split it asξ = (ξ− , ξ+) ∈ kZ− × kZ+, where Z− = {. . . ,−2,−1} andZ+ = {0,1, . . .}.

The first Oseledets direction only depends on the past, whilethe second one only depends on the future:

E1(ξ) = E1(ξ−), E2(ξ) = E2(ξ+).

Some facts about the sets of directions

C1 :=�

E1(ξ−); ξ ∈ kZ−

, C2 :=�

E2(ξ+); ξ ∈ kZ+

M is a multicone ⇒ C1 =∞⋂

i1,...,in

Ai1 · · ·Ain(M) (nested)

Proof: E1(ξ−) appears when we take ij = ξ−j.

forward non-overlapping condition (NOC+)

E1 : kZ− → C1 is a bijection.

C1 is a Cantor set.

Analogously for E2 : kZ+ → C1.

Some facts about the sets of directions

C1 :=�

E1(ξ−); ξ ∈ kZ−

, C2 :=�

E2(ξ+); ξ ∈ kZ+

M is a multicone ⇒ C1 =∞⋂

i1,...,in

Ai1 · · ·Ain(M) (nested)

Proof: E1(ξ−) appears when we take ij = ξ−j.

forward non-overlapping condition (NOC+)

E1 : kZ− → C1 is a bijection.

C1 is a Cantor set.

Analogously for E2 : kZ+ → C1.Jairo Bochi Optimization of Lyapunov Exponents 34 / 48

Main proposition

Proposition (E1 determines E2 a.s.)Suppose that the cocycle satisfies the forwardnon-overlapping condition (NOC+).Let μ be an ergodic measure such that λ1(μ) = λ⊤1 or λ⊥1.

Then for μ-a.e. ξ ∈ X, the direction E1(ξ−) is uniquelydetermined the direction E2(ξ+).

Main proposition

Proposition (E1 determines E2 a.s.)Suppose that the cocycle satisfies the forwardnon-overlapping condition (NOC+).Let μ be an ergodic measure such that λ1(μ) = λ⊤1 or λ⊥1.Then for μ-a.e. ξ ∈ X, the direction E1(ξ−) is uniquelydetermined the direction E2(ξ+).

Main Proposition ⇒ TheoremSuppose that the cocycle satisfies the non-overlappingcondition (NOC+ +NOC−). Let μ be an ergodic measuresupported in K⊤, that is, λ1(μ) = λ⊤1.

Then for μ-a.e. ξ ∈ X, the past ξ− uniquely the future ξ+:

ξ− ξ+

E1(ξ−) E2(ξ+)Proposition

NOC−

This implies that h(μ) = 0, for all μ supported in K⊤.

By the Entropy Variational Principle, htop(K⊤) = 0. Analogouslyfor K⊥.

This proves the Theorem modulo the Proposition.

ξ− ξ+

NOC−

This proves the Theorem modulo the Proposition.

ξ− ξ+

NOC−

This proves the Theorem modulo the Proposition.Jairo Bochi Optimization of Lyapunov Exponents 36 / 48

PROOF OF THE MAIN PROPOSITION

Cross ratio

Given nonzero vectors u, v, u′, v′ ∈ R2, (no three of thembeing collinear), we define their cross ratio:

[u,v;u′,v′] :=u× u′

u× v′·v × v′

v × u′∈ R ∪ {∞} ,

where × is the cross product on R2 (that is, determinant).

Actually the value above only depends on the directionsdetermined by the vectors. So we can define the cross ratio offour points in P1.

The cross ratio is invariant by linear isomorphisms.

Configurations of four points in P1

Let u, v, u′, v′ ∈ P1 be distinct. The configuration is calledco-parallel, crossing or anti-parallel according to the value ofthe cross ratio [u,v;u′,v′] as follows:

Co-parallel Crossing Anti-parallel

0 < [u,v;u′,v′] < 1 [u,v;u′,v′] > 1 [u,v;u′,v′] < 0

u′ u

v′ u

u′ u

Restrictions among the Oseledets directionsLemma (Geometrical Lemma)K⊤, K⊥ = Mather sets of an 1-step dominated cocycle.

Then ξ, η ∈ K⊤ ⇒�

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≥ 1 ,

ξ, η ∈ K⊥ ⇒�

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≤ 1 .

In particular:

co-parallel configuration crossing configurationis forbidden en K⊤: is forbidden en K⊥:

E1(η)E2(η)

E2(ξ) E1(ξ)

E1(η)E2(ξ)

E1(η) E1(ξ)

Restrictions among the Oseledets directionsLemma (Geometrical Lemma)K⊤, K⊥ = Mather sets of an 1-step dominated cocycle.

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≥ 1 ,

ξ, η ∈ K⊥ ⇒�

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≤ 1 .

In particular:co-parallel configuration crossing configuration

is forbidden en K⊤: is forbidden en K⊥:

E1(η)E2(η)

E2(ξ) E1(ξ)

E1(η)E2(ξ)

E1(η) E1(ξ)

Consequences of the Geometrical Lemma

Consider the set G :=�

(E1(ξ),E2(ξ)); ξ ∈ K⊤

⊂ P1 × P1.

Since the co-parallel configuration is forbidden, the set G hasa monotonicity property: if E1 moves clockwise then sodoes E2.

This implies that G is a graph (aboveits projection), with the exception toan at most countable numbers of“plateaux” and “cliffs”:

(E1(ξ),E2(ξ)); ξ ∈ K⊤

⊂ P1 × P1.

(E1(ξ),E2(ξ)); ξ ∈ K⊤

⊂ P1 × P1.

Geometrical Lemma ⇒ Main PropositionRecall the Main Proposition

PropositionNOC+ ⇒ E1 determines E2 almost surely with respect to anyλ1-maximizing or -minimizing measure.

Let μ be an ergodic non-periodic (and so non-atomic) measuresupported in K⊤. (The periodic case is trivial.)

Then the (countable) union of plateaux and cliffs in G is theimage under (E1,E2) of a set of zero μ-measure. (Here we needthe NOC+.)

Therefore each of the directions E1 and E2 uniquelydetermines the other μ-a.e., as we wanted to prove.

The case of K⊥ is similar (but extra care is needed:monotonicity is only local).

SummaryAs explained before, we obtain zero entropy since the pastξ− uniquely the future ξ+ μ-a.e.:

ξ− ξ+

NOC−

Barabanov functions

Geometrical Lemma (cross-ratios, forbidden configurations)

Main Proposition

THE END (?)Jairo Bochi Optimization of Lyapunov Exponents 43 / 48

PROOF OF THE GEOMETRICAL LEMMA

Some important ideas in the proof come from the 2002 paperby Bousch and Mairesse (counter-examples to the finitenessconjecture).

RecallLemma (Geometrical Lemma)K⊤, K⊥ = Mather sets of an 1-step dominated cocycle.

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≥ 1 ,

ξ, η ∈ K⊥ ⇒�

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≤ 1 .

In particular:

co-parallel configuration crossing configurationis forbidden en K⊤: is forbidden en K⊥:

E1(η)E2(η)

E2(ξ) E1(ξ)

E1(η)E2(ξ)

E1(η) E1(ξ)

RecallLemma (Geometrical Lemma)K⊤, K⊥ = Mather sets of an 1-step dominated cocycle.

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≥ 1 ,

ξ, η ∈ K⊥ ⇒�

�[E1(ξ),E1(η);E2(ξ),E2(η)]�

� ≤ 1 .

In particular:co-parallel configuration crossing configuration

is forbidden en K⊤: is forbidden en K⊥:

E1(η)E2(η)

E2(ξ) E1(ξ)

E1(η)E2(ξ)

E1(η) E1(ξ)

An important estimate

Let ξ ∈ K⊤, u ∈ E1(ξ). Let v ∈−→M be such that u− v ∈ E2(ξ).

Then |||u||| ≤ |||v|||.

E1(ξ)

E2(ξ)

v ∈−→M

Proof of the “important estimate” Lemma

E1(ξ)

E2(ξ)

v ∈−→M

Lemma: ξ ∈ K⊤ ⇒ |||u||| ≤ |||v|||

Proof: For all n ≥ 0,

un := A(n)(ξ)(u) ⇒ |||un||| = enλ⊤1 |||u|||

vn := A(n)(ξ)(v) ⇒ |||vn||| ≤ enλ⊤1 |||v|||

and so:|||v|||

|||u|||≥|||vn|||

|||un|||=: Qn

To prove the Lemma we show that Qn→ 0 as n→ +∞.

That’s easy, using that that ∠(un,vn)→ 0 exponentially fastand “Lipschitz property” of the Barabanov norms.

E1(ξ)

E2(ξ)

v ∈−→M

Lemma: ξ ∈ K⊤ ⇒ |||u||| ≤ |||v|||

un := A(n)(ξ)(u) ⇒ |||un||| = enλ⊤1 |||u|||

vn := A(n)(ξ)(v) ⇒ |||vn||| ≤ enλ⊤1 |||v|||

and so:|||v|||

|||u|||≥|||vn|||

|||un|||=: Qn

E1(ξ)

E2(ξ)

v ∈−→M

Lemma: ξ ∈ K⊤ ⇒ |||u||| ≤ |||v|||

un := A(n)(ξ)(u) ⇒ |||un||| = enλ⊤1 |||u|||

vn := A(n)(ξ)(v) ⇒ |||vn||| ≤ enλ⊤1 |||v|||

and so:|||v|||

|||u|||≥|||vn|||

|||un|||=: Qn

Proof of the Geometrical LemmaAssume for a contradiction that ξ, η ∈ K⊤ are in co-parallel

configuration: .

Choose a nonzero vector u ∈ E1(ξ), andlet v, w be as in the figure.

E2(ξ)

E1(ξ)E1(η)E2(η)

(Rem.: hidden cross-ratio)

Apply the “important estimate” twice:

|||u||| ≤ |||v||| ≤ |||w|||. Contradiction!

configuration: . Choose a nonzero vector u ∈ E1(ξ), andlet v, w be as in the figure.

E2(ξ)

E1(ξ)E1(η)E2(η)

configuration: . Choose a nonzero vector u ∈ E1(ξ), andlet v, w be as in the figure.

E2(ξ)

E1(ξ)E1(η)E2(η)

Optimization of Lyapunov Exponents

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