The entropy of Lyapunov-optimizing measures of some matrix cocycles

We consider one-step cocycles of $2 \times 2$ matrices, and we are interested in their Lyapunov-optimizing measures, i.e., invariant probability measures that maximize or minimize a Lyapunov exponent. If the cocycle is dominated, that is, the two Lyapunov exponents are uniformly separated along all orbits, then Lyapunov-optimizing measures always exist, and are characterized by their support. Under an additional hypothesis of nonoverlapping between the cones that characterize domination, we prove that the Lyapunov-optimizing measures have zero entropy. This conclusion certainly fails without the domination assumption, even for typical one-step $\mathrm{SL}(2,\mathbb{R})$-cocycles; indeed we show that in the latter case there are measures of positive entropy with zero Lyapunov exponent.


Introduction
Ergodic Optimization is concerned with the maximization or minimization of Birkhoff averages of a given function (called the potential ) over a given dynamical system: see [Je'06].A paradigm of this subject is that for sufficiently hyperbolic base dynamics and for typical potentials, optimizing orbits should have low dynamical complexity.This is confirmed in by a recent result by Contreras [Co], who showed that the optimizing orbits with respect to generic Lipschitz potentials over an expanding base are periodic.An important component of Contreras' proof is the fact previously shown by Morris [Mo'08] that in this generic situation, optimizing orbits have subexponential complexity (i.e., zero entropy).
In this paper we are interested in Ergodic Optimization in a noncommutative setting.We will replace Birkhoff sums by matrix products, and the quantities we want to maximize or minimize are the associated Lyapunov exponents.We would like to know whether the low complexity phenomena mentioned above is also typical in this noncommutative setting.
A natural starting point is to consider one-step matrix cocycles.In this case, the optimization problems above can be restated in more elementary terms: we are given finitely many square matrices, and we want to find sequences of products of these matrices attaining the maximum or minimum growth rate.These maximization and minimization problems were first considered by Rota and Strang [RS'60] and by Gurvits [Gu'95], respectively.The associated growth rates are called joint spectral radius and joint spectral subradius, respectively; they play an important role in Control Theory and there is a large body of literature about them (especially about the former): see the monograph [Ju'09] and references therein.An important contribution to this field was made by Bousch and Mairesse [BMa'02] Date: November, 2014 (this version).
1 who showed that the maximizing products are not always periodic, thus disproving the so called Finiteness Conjecture.(Of course, for one-step cocycles the parameter space is finite-dimensional and thus perturbative arguments are more difficult.) In this paper we deal with 2 ˆ2 one-step cocycles.We give explicit open conditions that ensure that the Lyapunov-optimizing orbits form a set of low complexity, more precisely of zero topological entropy.These conditions are related to hyperbolicity on the projective space, and are satisfied in some of the counterexamples to the Finiteness Conjecture exhibited in the literature.
In order to substantiate the importance of the hyperbolicity hypotheses, we also show that for typical non-hyperbolic one-step SLp2, Rq-cocycles the set of minimizing orbits has positive topological entropy.
Let us proceed with the precise definitions and results.
1.1.Extremal Lyapunov exponents for 2 ˆ2 matrix cocycles.Let Ω be a compact metric space and let T : Ω Ñ Ω be a continuous transformation.Let A : Ω Ñ GLp2, Rq be a continuous map.The pair pT, Aq is a called a 2 ˆ2 matrix cocycle.We are interested in the following products, the logarithms of which play the role of Birkhoff sums in our noncommmutative setting: A pnq pωq :" ApT n´1 ωq ¨¨¨Apωq, ω P Ω, n ě 0. (1.1) The Lyapunov exponents of the cocycle at a point ω P Ω, when they exist, are the limits: λ 1 pA, ωq :" lim where, for definiteness, }L} is the Euclidian operator norm of a matrix L, and mpLq :" }L ´1} ´1 is its mininorm.Let i P t1, 2u.If µ is a T -invariant probability measure then λ i pA, ωq exists for µ-almost every ω, we denote λ i pA, µq " ş λ i pA, ωq dµpωq.If µ is ergodic then λ i pA, ωq " λ i pA, µq for µ-almost every ω.
The maximal (or top) and minimal (or bottom) Lyapunov exponents are defined respectively as: where M T denotes the set of all T -invariant Borel probability measures.These four numbers are called the extremal Lyapunov exponents of the cocycle.A basic question is whether the sup's and inf's that appear in (1.3) are attained.The answer is "yes" in the cases of λ ⊤ 1 and λ ⊥ 2 , and "not necessarily" in the cases of λ ⊤ 2 and λ ⊥ 1 ; see subsection A.3.However, under the assumption of domination (that we will explain next), all sup's and inf's in (1.3) are attained.
1.2.Domination.Consider a 2 ˆ2 matrix cocycle pT, Aq where T is a homeomorphism.Suppose that for each ω P Ω we are given a splitting of R 2 as the sum of two one-dimensional subspaces e 1 pωq, e 2 pωq.We say that this is a dominated splitting with respect to the cocycle pT, Aq if the following properties hold: ‚ equivariance: Apωqpe i pωqq " e i pT ωq for all ω P Ω and i P t1, 2u; (1.4) ‚ dominance: there are constants c ą 0 and δ ą 0 such that }A pnq pωq|e 1 pωq} }A pnq pωq|e 2 pωq} ě ce δn for all ω P Ω and n ě 1. (1.5) An important property of dominated splittings is that they are always continuous, that is, e 1 and e 2 , viewed as maps from Ω to the projective space P1 , are automatically continuous (see e.g.[BDV'05, § B.1]).
We say that a cocycle is dominated if it admits a dominated splitting.Some authors say that the cocycle is exponentially separated, which is perhaps a better terminology.Domination is also sometimes called projective hyperbolicity, because it can be expressed in terms of uniform contraction and expansion on the projective space.
As shown in [Yo'04, BG'09], a 2 ˆ2 cocycle pT, Aq is dominated if and only if there are constants c ą 0 and δ ą 0 such that }A pnq pωq} mpA pnq pωqq ě ce δn for all ω P Ω and n ě 0. (1.6) Note that the LHS is a measure of "non-conformality" of the matrix A pnq pωq.
If a cocycle is dominated then the Lyapunov exponents (1.2) are always distinct; actually δ ą 0 as in (1.6) is a uniform lower bound for the gap between them.Moreover (see subsection 2.1) the Lyapunov exponents are given by integrals: As a consequence of these formulas, the problem of maximizing or minimizing Lyapunov exponents for dominated cocycles is equivalent to the optimization of Birkhoff averages of the continuous functions ϕ i , and so many standard results apply (see [Je'06]). 1 In particular, one can easily show that: Another consequence is a fact we mentioned already: all sup's and inf's that appear in (1.3) are attained in the dominated case.
1.3.One-step cocycles.Fix an integer k ě 2. Let Ω " t1, . . ., ku Z be the space of bi-infinite words on k symbols.With some abuse of notation, we denote this set by k Z .Let T : k Z Ñ k Z be the shift transformation.Given a k-tuple of matrices A " pA 1 , . . ., A k q P GLp2, Rq k , we associate with it the locally constant map A : k Z Ñ GLpd, Rq given by Apωq " A ω0 .In this case, pT, Aq is called a one-step cocycle, and the products (1.1) are simply The k-tuple of matrices A is called the generator of the cocycle.We denote λ ⊤ i pAq :" λ ⊤ i pAq and λ ⊥ i pAq :" λ ⊥ i pAq.
considers regular points, as the alert reader have noticed.On the other hand, non-regular points have no effect in the optimization of Birkhoff averages (see [Je'06]).Therefore, for dominated cocycles at least, non-regular points have no effect in the optimization of Lyapunov exponents.
We remark that for one-step cocycles, the values λ ⊤ 1 pAq and λ ⊥ 1 pAq can be alternatively defined in a more elementary way (without speaking of measures) as: The numbers ̺ ⊤ pAq :" e λ ⊤ 1 pAq and ̺ ⊥ pAq :" e λ ⊥ 1 pAq are called joint spectral radius and joint spectral subradius and constitute an active topic of research: see [Ju'09].Let us remark that that the joint spectral radius is always a continuous function of the matrices, while the continuity of the joint spectral subradius is far from automatic and is related to domination: see [BMo].
1.4.Domination for one-step GLp2, Rq cocycles.A one-step cocycle pT, Aq is dominated if and only if the number pλ 1 ´λ2 q ⊥ pAq :" inf µPMT pλ 1 pA, µq ´λ2 pA, µqq (1.10) is positive; see subsection A.2 for the (easy) proof.Let us see still another characterization of domination for one-step cocycles. The ˚is an image of C `by a linear isomorphism.A multicone in R 2 ˚is a disjoint union of finitely many cones.
We say that a multicone M Ă R 2 ˚is forward-invariant with respect to A " pA 1 , . . ., A k q if the image multicone Ť i A i pM q is contained in the interior of M .For example, if the A i 's have positive entries then the standard positive cone C ìs a forward-invariant multicone for pA 1 , . . ., A k q.For more complicate examples, see [ABY'10].
It was proved in [ABY'10, BG'09] that the one-step cocycle generated by A is dominated if and only if A has a forward-invariant multicone.
If M is a multicone, its complementary multicone M co is defined as the closure Notice that if M is forward-invariant with respect to pA 1 , . . ., A k q then M co is backwards-invariant, that is, forward-invariant with respect to pA ´1 1 , . . ., A ´1 k q.
1.5.Mather sets.Under the assumptions above, the extremal Lyapunov exponents "live" in certain invariant sets: Theorem 1. Suppose that the one-step cocycle generated by A P GLp2, Rq k is dominated.For each ‹ P t⊤, ⊥u, let K ‹ be the union of all supports of measures µ P M T such that λpµq " λ ‹ 1 .Then: An obvious consequence of the theorem is the existence of λ 1 -optimizing measures.
We call K ⊤ and K ⊥ upper and lower Mather sets, respectively.Our upper Mather set corresponds to what Morris [Mo'13] calls a Mather set.The terminology is coherent with Lagrangian Dynamics, where Mather sets were first studied in [Ma'91].The existence of Mather sets is related to the Bousch's "subordination principle": see [Mo'07] for the commutative context, and [CZ'13] for the subadditive context.
Actually the existence of the upper Mather set is guaranteed for 1-step cocycles (in any dimension) without assumptions of domination: see [Mo'13].
The existence of both Mather sets in Theorem 1 can be deduced from Hölder continuity of the Oseledets directions using the usual (commutative) ergodic optimization theory.However, the proof of Theorem 1 that we will present is selfcontained and gives extra information which will be useful in the proof of our major result, Theorem 2 below.1.6.Zero entropy.We say that A " pA 1 , . . ., A k q satisfies the forward NOC (nonoverlapping condition) if it has a forward-invariant multicone M Ă R 2 ˚such that A i pM q X A j pM q " ∅ whenever i ‰ j .
We say that A " pA 1 , . . ., A k q satisfies the backwards NOC if pA ´1 1 , . . ., A ´1 k q satisfies the forward NOC.We say that pA 1 , . . ., A k q satisfies the NOC if it satisfies both the forward and the backwards NOC.Remark 1.1.The forward and the backwards NOC are not equivalent: for example, if then pA 1 , A 2 q satisfies the forward NOC, but not the backwards NOC, as one can easily check.Ÿ The main result of this paper is the following: Theorem 2. For every k and every A P GLp2, Rq k , if the one-step cocycle generated by A is dominated and satisfies the NOC then the restriction of the shift map to either Mather set K ⊤ or K ⊥ has zero topological entropy.
The conclusion of the theorem means that for each ‹ P t⊤, ⊥u, the number w ‹ pℓq of words of length ℓ in the alphabet t1, . . ., ku that can be extended to a bi-infinite word in the Mather set K ‹ is a subexponential function of ℓ, that is, lim ℓÑ8 1 ℓ log w ‹ pℓq " 0. (1.11) (see [Pe'89,).
There are examples where Theorem 2 applies and K ⊤ is non-discrete: In the family of examples given in [BMa'02] where a maximizing measure is Sturmian non-periodic, the NOC condition holds for some choices of the parameters.
There are also examples where Theorem 2 applies and either K ⊤ or K ⊥ is not uniquely ergodic: see subsection A.4. 1.7.Positive entropy.As a counterpoint to Theorem 2, we will exhibit nontrivial situations where λ 1 -minimizing measures with positive entropy exist.
A cocycle pT, Aq is called uniformly hyperbolic if it has an equivariant splitting into two subbundles, one being uniformly expanding and the other being uniformly contracting.Any uniformly hyperbolic cocycle is dominated, and the converse holds for SLp2, Rq-cocycles.
Theorem 3. Fix k ě 2 and let T be the full shift in k symbols.There exists an open and dense subset U of SLp2, Rq k such that for every A P U, i) either the one-step cocycle over T generated by A is uniformly hyperbolic; ii) or there exists a compact T -invariant set K Ă k Z of positive topological entropy and such that the norms }A pnq pωq} are uniformly bounded over pω, nq P K ˆZ.
Notice that in the first case we have λ ⊥ 1 pAq ą 0, while in the second case it follows from the entropy variational principle (see [Pe'89,p. 269]) that there exists a measure µ P M T such that h µ pT q ą 0 and λ 1 pA, µq " 0.
1.8.Organization of the paper and overview of the proofs.In section 2 we collect basic fact about dominated cocycles.
A standard procedure to solve ergodic optimization problems is to look for a change of variables under which the optimizing orbits become evident, or "revealed".Following this idea, in section 3 we construct what we call "Barabanov functions" (in analogy to the Barabanov norms from joint spectral radius theory), and immediately use them to prove the existence of the Mather sets (Theorem 1).
In section 4 we use the Barabanov functions to prove that the directions of the dominated splitting for points on the Mather sets must obey severe geometrical obstructions, which in turn imply that one direction uniquely determines the other, with an at most countable number of exceptions.Using this property, we prove Theorem 2 in section 5.
Let us mention that some of the key parts in sections 3 and 4 were inspired by ideas from the paper of Bousch and Mairesee [BMa'02].Nevertheless, we do not use their results directly.
The simpler proof of Theorem 3 is given in section 6, and is independent of the previous sections.
In Appendix A we present complementary information, including counterexamples showing the limits of our results and alternative definitions for some of the concepts we have discussed.In the final subsection A.5, we pose a few problems and suggest some directions for future research.

Preliminaries: Basic facts about 2 ˆ2 dominated cocycles
In this section we collect some simple facts about dominated cocycles that will be needed in the sequel.Even though these facts are standard, for the convenience of the reader we will provide some of the proofs.
2.1.General cocycles.In this subsection, let Ω be a compact metric space, let T : Ω Ñ Ω be a homeomorphism, let A : Ω Ñ GLp2, Rq be a continuous map, and assume that the cocycle pT, Aq has a dominated splitting into directions e 1 , e 2 .
Lemma 2.1.If ω P Ω and x P R 2 e 2 pωq then Proof.Easy and left to the reader.
Proof.Let f 1 ' f 2 be another dominated splitting for the cocycle pT, Aq.Fix ω P Ω and consider the product Then the first and the third factors tend to 8 as n Ñ 8.If f 2 pωq ‰ e 2 pωq then by Lemma 2.1 the second and the fourth factors have nonzero finite limits as n Ñ 8, which is a contraction.This shows that f 2 " e 2 .The same reasoning for the inverse cocycle shows that f 1 " e 1 .
Proof.This is a consequence of the fact that the angles between the directions of the dominated splitting are uniformly bounded from below.The details are left to the reader.
It follows that the Lyapunov exponents defined by (1.2) can be determined from the restriction of the cocycle to the directions that form the dominated splitting: Corollary 2.4.If the cocycle pT, Aq is dominated then, for any i P t1, 2u, for every ω P Ω such that at least one of these quantities is well-defined.
Notice that the RHS of (2.2) is a limit of Birkhoff averages, so the integral formulas (1.7) follow.Remark 2.5.Actually Lemma 2.3 implies that the dominated splitting coincides with the Oseledets splitting wherever the latter is defined.The properties expressed by Corollary 2.4 and formulas (1.7) hold in general for Oseledets splittings.Ÿ 2.2.One-step cocycles.Let us fix some notation.The projective space of R 2 is denoted by ˚is a multicone then let M 1 :" x 1 P P 1 ; x P M ( .The following result provides an useful "adapted metric" on the multicone.Similar constructions appear in [ABY'10,BG'09,BMo].
Lemma 2.6.Assume that pA 1 , . . ., A k q P GLp2, Rq k generates a dominated onestep cocycle, and let M Ă R 2 be a forward-invariant multicone.There exist a metric d on the projectivization M 1 and constants c 1 ą 1 and 0 ă τ ă 1 such that for all x, y P M , we have (2.4) Proof.By a compactness argument, there exists an open neighborhood U of M 1 in P 1 such that A 1 i pU q Ă M 1 for all i P t1, . . ., ku.We can assume that each connected component of U contains exactly one connected component of M 1 .
Endow each connected component of U with its Hilbert metric, and restrict it to the corresponding connected component of M 1 .We use the same letter d to denote all those metrics.Rescaling if necessary, we can assume that d ď 1{2 whenever defined.Moreover, there are constants c 1 ą 1 and 0 ă τ ă 1 such that properties (2.3) and (2.4) hold whenever x 1 and y 1 are in the same connected component of M 1 .
Given x 1 , y 1 P M 1 , define ℓpx 1 , y 1 q as the least integer n ě 0 with the property that for all ω P k Z , the directions A pnq pωq 1 x 1 and A pnq pωq 1 y 1 belong to the same connected component of M 1 .Notice that such a number always exists and is not bigger than the square of the number of components of M 1 ; this follows by a pigeonhole argument using the fact that no linearly-induced map can have two disjoint strictly invariant intervals.
The function ℓ has the following property: and satisfies an ultrametric inequality: We now extend d by setting d px 1 , y 1 q :" ℓ px 1 , y 1 q if x 1 and y 1 are in different connected components of M 1 .Then d is a distance function.Moreover, increasing c 1 and τ if necessary, properties (2.3) and (2.4) are satisfied.
Proposition 2.7.Assume that pA 1 , . . ., A k q P GLp2, Rq k generates a dominated one-step cocycle, Let e 1 , e 2 : k Z Ñ P 1 be the invariant directions forming the dominated splitting, and let M Ă R 2 ˚be a forward-invariant multicone, and let M co be the (backwards-invariant) complementary multicone.Then for any ω P k Z we have To prove this proposition, we will use Lemma 2.6 and the following fact: Lemma 2.8.
Proof.Up to an error op}x ´y}q, the triangle with vertices 0, x, y has area }x} 2 >px, yq, while the triangle with vertices 0, Bx, By has area }Bx} 2 >pBx, Byq.
Since the linear map B expands areas by a factor | det B|, the lemma follows.
Proof of Proposition 2.7.Assume given a multicone M for pA 1 , . . ., A k q.We will show that formulas (2.5), (2.6) define directions forming a dominated splitting; therefore the proposition will follow from the uniqueness of dominated splittings (Proposition 2.2).
Indeed, for every ω P k Z , the RHS of (2.5) is a nested intersection of compact subsets of P 1 whose diameters, by Lemma 2.6, converge to 0. Therefore the intersections contains a single point in P 1 ; call it e 1 pωq.Analogously we define e 2 pωq using (2.6).This pair of directions forms a continuous splitting with the equivariance property (1.4).Let us check that this splitting is dominated.By Lemma 2.6, there exist constants c 1 ą 1, 0 ă τ ă 1 such that > `Apnq pωqx, A pnq pωqy >px, yq ď c 2 1 τ n for all ω P k Z , x, y P M, n ě 0 .
Taking x P e 1 pωq and using Lemma 2.8, we conclude that By an analogous argument, there exist constants c1 ą 1, 0 ă τ ă 1 such that Therefore the domination property (1.5) holds for appropriate constants c, δ.As explained before, the proposition follows from uniqueness of dominated splittings. 2 Let Z ´(resp.Z `) be the set of negative (resp.nonnegative) integers.Define the following projections: π ´pωq " p. . ., ω ´2, ω ´1q , (2.7) As a straightforward consequence of Proposition 2.7, we have: Corollary 2.9.Assume that pA 1 , . . ., A k q P GLp2, Rq k generates a dominated onestep cocycle.Then, for each ω P k Z , the invariant directions e 1 pωq and e 2 pωq forming the dominated splitting depend only on π ´pωq and π `pωq, respectively, and therefore there exist continuous maps ẽ1 , ẽ2 such that the following diagrams commute: (2.9) Moreover: ‚ If the forward NOC is satisfied then ẽ1 is one-to-one; in particular e 1 pk Z q is a Cantor set.‚ If the backwards NOC is satisfied then ẽ2 is one-to-one; in particular e 2 pk Z q is a Cantor set.
2 Incidentally, we have just proved that the existence of a forward-invariant multicone implies Unfortunately, no such norm can in general exist, even assuming irreducibility of A. For example, if the cocycle is such that λ ⊤ 2 pAq ă λ ⊥ 1 pAq then applying relation (3.2) to the orbit of a nonzero vector in the second Oseledets direction e 2 we reach a contradiction.
This example shows that if such a "minimizer Barabanov norm" exists, relation (3.2) cannot hold for all vectors, but only for vectors away from the e 2 -directions.In general, the set of e 2 -directions can be large or even the whole P 1 , but for dominated cocycles it is a proper compact subset of P 1 .
As we show in this section, under the assumption of domination it is indeed possible to construct an object that retains the most useful properties of (the logarithm of) a "minimizer Barabanov norm".For convenience, we simultaneously consider both the maximizer and minimizer cases: Theorem 3.1.Assume that pA 1 , . . ., A k q P GLp2, Rq k generates a dominated onestep cocycle, and let M Ă R 2 be a forward-invariant multicone.Then there exist functions p ⊤ : M Ñ R and p ⊥ : M Ñ R with the following properties: ‚ extremality: for all x P M , max iPt1,...,ku Our construction combines their techniques with properties of multicones and the Hilbert metric.A higher-dimensional version of our construction was obtained in [BMo].
Let us also mention that similar constructions also play an important role on ergodic optimization, action minimization in Lagrangian dynamics, and optimal control: see [BMa'02] and references therein.
Back to the context of products of matrices, Morris [Mo'13] has studied relations between the Barabanov norm (3.1) and the upper Mather set.
In this paper, the upper and lower Barabonov functions are the main tools in the analysis of the upper and lower Mather sets.
3.2.Proofs.In the following proof of Theorem 3.1, we will also establish some facts that are necessary for the subsequent proof of Theorem 1.
Proof of Theorem 3.1.For each i P t1, . . ., ku, define h i : P 1 Ñ R by where, as before, a prime denoted projectivization and }¨} denotes the Euclidian metric.Fix a constant c 2 ą 0 such that ˇˇh i px 1 q ´hi py 1 q ˇˇď c 2 >px, yq for all x, y P R 2 ˚.
Let M be a forward-invariant multicone for pA 1 , . . ., A k q, and let d be the metric on the projectivization M 1 given by Lemma 2.6.Let B be the vector space of continuous functions from M 1 to R, endowed with the uniform (supremum) distance |¨| 8 .Let c 3 :" c 1 c 2 {p1 ´τ q and let K Ă B be the set of functions that are c 3 -Lipschitz with respect to d.
For each function f P K, define two functions T ‹ f : M 1 Ñ R (where ‹ P t⊤, ⊥u) by We claim that T ‹ f P K. Indeed, for all x 1 , y 1 P M 1 , we have Thus we have defined maps T ‹ : K Ñ K. Next, we claim that these maps are continuous.Indeed, for all f , g P K, we have Let B be the quotient of the space B by the subspace of constant functions; it is a Banach space endowed with the quotient norm | f | 8 :" inft|f | 8 ; πpf q " f u, where π : B Ñ B denotes the quotient projection.By the Arzelà-Ascoli theorem, the convex set K :" πpKq is compact.Since T ‹ commutes with the addition of a constant, there exists a map T ‹ : K Ñ K such that π ˝T ‹ " T ‹ ˝π.The map T ‹ is continuous, as it is easy to check; in particular, by the Schauder theorem, it has a fixed point f ‹ 0 .This means that there exist f ‹ 0 P K and p ‹ pxq :" f ‹ 0 px 1 q `log }x} for all x P M. Note that for every x P M , the following properties hold: property (3.5), max ˘.Taking c 0 " c 1 c 3 , we see that property (3.6) holds when x 1 and y 1 are in the same connected component of M 1 .Since the angle between directions in different components is uniformly bounded from below, we can increase c 0 if necessary so that property (3.6) fully holds.
To complete the proof of Theorem 3.1 we need to show that the numbers β ⊤ and β ⊥ that appear in (3.7) and (3.8) are respectively equal to the numbers λ ⊤ 1 and λ ⊥ 1 that appear in (3.3) and (3.4).As we prove these equalities, we will also establish some facts that will be useful in the forthcoming proof of Theorem 1.
(3.10)By Proposition 2.7 we have e 1 pωq Ă M , so the expression above makes sense because, and by (3.5) it does not depend on the choice of x; thus ψ ‹ is well-defined.Moreover it is a continuous function.By equivariance of the e 1 direction, for every ω P k Z , x P e 1 pωq t0u, and n ě 1 we have p ‹ pA pnq pωqxq ´p‹ pxq " n´1 ÿ j"0 ψ ‹ pT j ωq.
We will show that L ‹ is compact and nonempty.We begin by showing the following: Claim 3.2.For any ω ´P k Z´t here exists ω `P k Z`s uch that if ω " ω ´ω`i s concatenation of ω ´and ω `then ψ ‹ pT n ωq " β ‹ for all n ě 0.
Let L ‹ `be the set of ω P k Z such that ψ ‹ pT n ωq " β ‹ for all n ě 0, which by Claim 3.2 is nonempty.Since L ‹ `is compact and contains T pL ‹ `q, the set L ‹ " Ş ně0 T n pL ‹ `q is compact and nonempty, as announced.In particular, there exists at least one T -invariant probability measure µ ‹ supported on L ‹ , and so with λ 1 pµ ‹ q " β ‹ .Together with (3.11) this implies that β ‹ " λ ‹ 1 .So (3.3) and (3.4) respectively follow from (3.7) and (3.8) and the proof of Theorem 3.1 is complete.
Proof of Theorem 1.For each ‹ P t⊤, ⊥u, let M ‹ T be the set of measures µ P M T such that λpµq " λ ‹ 1 .We have seen in the proof of Theorem 3.1 that there exists a nonempty compact T -invariant set L ‹ such that µ P M ‹ T if and only if supp µ Ă L ‹ .Define the Mather set K ‹ as the union of the supports of all measures µ in M ‹ T , so To show that K ‹ is a compact set, we follow an argument from [Mo'13].The set of all Borel probabilities on k Z with the usual weak-star topology is metrizable and compact, and M T is a compact subset.Since L ‹ is compact, using Urysohn's lemma we see that the set M ‹ T is also compact.In particular, it has a countable dense sequence pν ‹ n q.Consider ν ‹ :" ř 2 ´nν ‹ n , which is an element of M ‹ T .It is then easy to show that supp ν ‹ " K ‹ , which in particular shows that K ‹ is compact.
The remaining assertions in Theorem 1 are now obvious, and the proof is complete.

Properties of Lyapunov-optimal orbits
In this section we explore consequences of Theorem 3.1.Let us remark that the results of this section do not require the nonoverlapping condition.
Fix a dominated one-step cocycle with generator pA 1 , . . ., A k q, a forward-invariant multicone M , and Barabanov functions p ⊤ , p ⊥ on M .4.1.Geometrical obstructions.In this subsection, we will show that the invariant directions of points on the Mather sets must obey certain geometrical obstructions.
Since the functions p ‹ are continuous, these sets are closed.
Notice that, as a consequence of properties (3.3) and (3.4) of the Barabanov functions, the following holds: Lemma 4.1.If pω `, xq P J ‹ and y P M are such that x ´y P ẽ2 pω `q (see Fig. 1) then: Proof.Let ω `P k Z`a nd x, y P M be such that x´y P ẽ2 pω `q.Let x n :" A pnq pω `qx and y n :" A pnq pω `qy, for n ě 0. Since x, y R ẽ2 pω `q, it follows from Lemma 2.1 that the quantities }x n ´yn } }x n } , }x n ´yn } }y n } , and >px n , y n q tend to 0 as n Ñ 8. (4.1) Let us now show that lim nÑ8 " p ‹ py n q ´p‹ px n q ‰ " 0 .(4.2) Indeed, by property (3.6) of Barabanov functions, ˇˇp ‹ py n q ´p‹ px n q ˇˇď c 0 >py n , x n q `ˇl og }y n } ´log }x n } ˇˇ.
The first term tends to zero as n Ñ 8.The second term can be estimated as: which by (4.1) tends to zero as well.This proves (4.2).Next, assume pω `, xq P J ‹ .So, for all n ě 0, By properties (3.3) and (3.4) we have: In particular, p ⊤ py n q ´p⊤ px n q ď p ⊤ pyq ´p⊤ pxq if ‹ " ⊤, p ⊥ py n q ´p⊥ px n q ě p ⊥ pyq ´p⊥ pxq if ‹ " ⊥.
Taking limits as n Ñ 8 and recalling (4.2) we obtain the lemma.
Given vectors x 1 , y 1 , x 2 , y 2 P R 2 ˚, no three of them collinear, we define their cross-ratio rx 1 , y 1 ; x 2 , y 2 s :" where ˆdenotes cross-product in R 2 , i.e. determinant.See [BK'53,Section 6].The cross-ratio actually depends only on the directions defined by the four vectors, which allows us to apply the same definition to 4-tuples in pP 1 q 4 without three coinciding points.Moreover, the cross-ratio is invariant under linear transformations.
We now use Lemma 4.1 to prove the following important Lemma 4.2, whose character is similar to Proposition 2.6 from [BMa'02]: Lemma 4.2.For all pξ, x 1 q, pη, y 1 q P J ‹ and nonzero vectors x 2 P ẽ2 pξq, y 2 P ẽ2 pηq we have Proof.Let us consider the case of J ⊤ ; the other case is analogous.
Recall from Proposition 2.7 that any e 1 direction is different from any e 2 direction.So, neither x 1 nor y 1 can be collinear to x 2 or y 2 .Hence the cross-ratio is well defined.Moreover, we can write: x 1 " αx 2 `βy 1 and y 1 " γy 2 `δx 1 .
In the rest of the section we will use the hyperbolic geometry representation of the projective space P 1 .Consider the unit disk D :" tz P C; |z| ă 1u endowed with the Poincaré hyperbolic metric.Given two different points x 1 , x 2 in the unit circle BD, let ÝÝÑ x 2 x 1 denote the oriented hyperbolic geodesic from x 2 to x 1 .We identify BD with the projective space P 1 as follows: e 2θi P BD Ø pcos θ, sin θq 1 P P 1 .
Let px 1 , y 1 ; x 2 , y 2 q be a 4-tuple of distinct points in P 1 .Then one and only one of the following possibilities holds: ‚ antiparallel configuration: x 1 ă y 2 ă y 1 ă x 2 ă x 1 for some cyclic order ă on P 1 (see Fig. 2); ‚ coparallel configuration: x 1 ă y 1 ă y 2 ă x 2 ă x 1 for some cyclic order ă on P 1 (see Fig. 3); ‚ crossing configuration: x 1 ă y 1 ă x 2 ă y 2 ă x 1 for some cyclic order ă on P 1 (see Fig. 4).We say that two geodesics ÝÝÑ x 2 x 1 and Ý Ý Ñ y 2 y 1 with distinct endpoints are antiparallel, coparallel, or crossing according to the configuration of the 4-tuple px 1 , y 1 ; x 2 , y 2 q.The configuration is expressed in terms of the cross-ratio as follows: Proposition 4.4.Consider a 4-tuple px 1 , y 1 ; x 2 , y 2 q of distinct points in P 1 .Then: Proof.With a linear change of coordinates, we can assume that the directions y 1 , x 2 , y 2 contain the vectors p1, 1q, p1, 0q, p0, 1q, respectively.Let pa, bq be a nonzero vector in the x 1 direction.Then rx 1 , y 1 ; x 2 , y 2 s " b{a.The proposition follows by inspection.
Define the following compact subsets of the torus P 1 ˆP1 : As a consequence of Lemma 4.3 and Proposition 4.4, together with the description of the Mather sets given by (3.12), we have: Corollary 4.5.Let px 1 , x 2 q, py 1 , y 2 q P G ‹ .Then: ‚ if ‹ " ⊤ then px 1 , y 1 ; x 2 , y 2 q cannot be in coparallel configuration; ‚ if ‹ " ⊥ then px 1 , y 1 ; x 2 , y 2 q cannot be in crossing configuration.4.2.Each invariant direction essentially determines the other.Now we will show that for points ω on the Mather sets, each invariant direction e 1 pωq or e 2 pωq uniquely determines the other, except for a countable number of bad directions.This fact (stated precisely in Lemma 4.6 below) is actually a simple consequence of Corollary 4.5, and forms the core of the proof of Theorem 2.
For each x P N ‹ 1 , let I ‹ pxq be the least closed subinterval of P 1 txu containing G ‹ 2 pxq.
We begin with the case of N ⊤ 1 .Claim 4. Proof of the claim.Let v and w be the endpoints of the interval I ⊤ pxq and take any point z in its interior.Then the geodesic Ý Ñ zy is coparallel to one of the two geodesics Ý Ñ vx or Ý Ñ wx.Since px, vq and px, wq belong to G ⊤ , by Corollary 4.5 we conclude that py, zq does not.This shows that G ⊤ 2 pyq X int I ⊤ pxq " ∅, and, in particular, BI ⊤ pyq X int I ⊤ pxq " ∅.An analogous argument gives BI ⊤ pxq X int I ⊤ pyq " ∅.It follows that int I ⊤ pxq X int I ⊤ pyq " ∅.
It follows from separability of the circle that N ⊤ 1 is countable.Now let us consider the case of N ⊥ 1 .For each x P N ⊥ 1 , let ∆pxq be the ideal triangle whose vertices are x and the two endpoints of the interval I ⊥ pxq.Proof of the claim.Let v and w be the endpoints of the interval I ⊤ pxq.Since these points belong to e 2 pK ⊥ q, which is disjoint from e 1 pK ⊥ q, none of them can be equal to y.Let C be the connected component of D int ∆pxq whose closure at infinity contains y.Let z P G ⊥ 2 pyq.By Corollary 4.5, the geodesic Ý Ñ zy does not cross Ý Ñ vx nor Ý Ñ wx.It follows that Ý Ñ zy is disjoint from int ∆pxq, and so it is contained in C. Since C is geodesically convex, it follows that ∆pyq Ă C.This proves the claim.
It follows from separability of the disc that N ⊥ 1 is also countable, thus completing the proof of Lemma 4.6.

Obtaining zero entropy
In this section we conclude the proof of Theorem 2. Fix a dominated one-step cocycle with generator pA 1 , . . ., A k q, and fix ‹ P t⊤, ⊥u.Recall the definition (4.4) of the set N ‹ 1 .Lemma 5.1.There exists a Borel measurable map g ‹ : e 1 pk Z q Ñ e 2 pk Z q such that ω P K ‹ e 1 pωq R N ‹ 1 * ñ g ‹ pe 1 pωqq " e 2 pωq . (5.1) Proof.Let x 1 P e 1 pk Z q.If x 1 P e 1 pK ‹ q N ‹ 1 then there exists a unique x 2 P e 2 pK ‹ q such that px 1 , x 2 q P G ‹ ; let g ‹ px 1 q :" x 2 .Otherwise if x 1 R e 1 pK ‹ q N ‹ 1 then let g ‹ px 1 q :" p, where p P e 2 pk Z q is an arbitrary constant.Thus we have defined a map g ‹ that satisfies property (5.1).Its graph is and therefore is a Borel measurable subset of e 1 pk Z qˆe 2 pk Z q.It follows (use [Bog'08, Lemma 6.7.1]) that g ‹ is a Borel measurable map, as we wanted to show.
By the entropy variational principle (see [Pe'89,p. 269]), in order to prove that the restriction of T to the compact invariant set K ‹ has zero topological entropy, it is sufficient to prove that h µ pT q " 0 for every ergodic probability measure µ supported on K ‹ .Fix any such measure µ.Let us assume that µ is non-atomic, because otherwise there is nothing to prove.
Lemma 5.2.µ `e´1 1 pN ‹ 1 q ˘" 0. Proof.By Lemma 4.6, the set N ‹ 1 Ă P 1 is countable.Since A has the forward NOC, it follows from Corollary 2.9 that the set e ´1 1 pN ‹ 1 q Ă k Z is a countable union of sets of the form tω ´u ˆkZ`.Assume for a contradiction that e ´1 1 pN ‹ 1 q has positive measure.Then there exists ω ´P k Z´s uch that F :" tω ´u ˆkZ`h as positive measure.By Poincaré recurrence, there exists p ě 1 such that T ´ppF q X F ‰ ∅.
It follows that the infinite word ω ´is periodic with period p, which in turn implies that T ´ppF q Ă F .By invariance, µ pF T ´ppF qq " 0 and µ ˜č ně0 T ´np pF q ¸" µpF q ´µ `F T ´ppF q ˘´µ `T ´ppF q T ´2p pF q ˘´¨¨" µpF q ą 0.
But the set Ş ně0 T ´np pF q is a singleton, thus contradicting the assumption that µ is non-atomic.This proves the lemma.
As an immediate consequence of the previous two lemmas, we obtain: e 2 " g ‹ ˝e1 µ-a.e. (5.2) That is, the direction e 1 almost surely determines e 2 .Consider the continuous maps ẽ1 , ẽ2 given by Corollary 2.9.Due to the backwards NOC, the map ẽ2 is one-to-one, and therefore there exists a unique map f ‹ that makes the following diagram commutative: Moreover, the map f ‹ is Borel measurable, and by the commutativity relations (2.9) and ( 5.2), it satisfies: (5.3) That is, the past almost surely determines the future.It is known that this property implies zero entropy, but for the reader's convenience let us spell out the details.Let C " tC 1 , . . ., C k u be the partition of k Z into cylinders Define another partition C " t C1 , . . ., Ck u by Cj :" π ´1 ´`pf ‹ q ´1 pπ `pC j qq ˘.
These are Borel sets that actually belong to the the following σ-algebra: By (5.3), µpC j △ Cj q " 0, that is C " C modulo zero sets.So, using the Kolmogorov-Sinai theorem and other basic facts about entropy (see e.g.[Pe'89]), we obtain: This proves Theorem 2.

Obtaining positive entropy
In this section we prove Theorem 3.
6.1.Sufficient conditions for the existence of many bounded products.
Lemma 6.1.Given a sequence B 0 , B 1 , . . . of matrices in SLp2, Rq, let P i :" B i´1 ¨¨¨B 0 and let u i , v i be unit vectors in R 2 such that P i u i " }P i }v i .Suppose that there are constants 0 ă κ ă 1 ă C such that }B i } ď C and }B i v i } ď κ for every i. Then 1 ´κ2 for every i.Proof.Recall that the Hilbert-Schmidt norm of a matrix A is defined as }A} HS :" ?tr A ˚A.If A P SLp2, Rq then }A} 2 HS " }A} 2 `}A} ´2.Let B i , P i , u i , v i , C and κ be as in the statement of the lemma.Let v K i be a unit vector orthogonal to v i .With respect to the basis tv i , v K i u we can write HS ď 2C 2 {p1 ´κ2 q for every i, which implies the lemma.
Given A " pA 1 , . . ., A k q P SLp2, Rq k , let xAy be the semigroup generated by A, that is, the set of all products of the form A in . . .A i1 (where n ě 1).
Let C be the set of A P SLp2, Rq k such that for every v P R 2 and ε ą 0 there exists P P xAy such that }P v} ă ε.It is easily seen that A P C if and only if for every unit vector v P S 1 there exists P " A i npvq ¨¨¨A i1 P xAy such that }P v} ă 1.It follows from compactness of the unit circle that the lengths npvq can be chosen uniformly bounded, and that C is open.Lemma 6.2.Every A P C satisfies the second alternative in Theorem 3.
Proof.Fix A P C. Let C :" max }A i }.It is an easy exercise to show that there exist κ P p0, 1q and an integer ℓ ě 2 such that for every unit vector v P R 2 there exists a product P P xAy of length ℓ ´1 such that }P v} ă C ´1κ, and in particular }A i P v} ă κ for every i " 1, . . ., k.Let L :" ω P k Z ; }A ℓn pωq} ď C 1 @n P Z ( , where C 1 :" ? 2 C ℓ ? 1 ´κ2 It follows from Lemma 6.1 that L ‰ ∅; actually given any bi-infinite sequence of symbols . . ., ω ´ℓ, ω 0 , ω ℓ , ω 2ℓ , . . . in the alphabet t1, . . ., ku, we can choose the remaining symbols to form a word ω in L.

Let
K :" ω P k Z ; }A n pT m ωq} ď C 2ℓ C 2 1 @n, m P Z ( .By definition, this set is compact and T -invariant, and it is easy to see that it contains L. It follows from the previous observations about L that the topological entropy of K is at least ℓ ´1 log k, and thus positive as required.
6.2.Checking denseness.Let H be the set of A P SLp2, Rq k such that the onestep cocycle generated by A is uniformly hyperbolic.Consider the open set U :" H Y C. By Lemma 6.2, every element of U satisfies one of the alternatives of Theorem 3. Therefore, to prove the theorem, it is sufficient to show that U is dense.
Let E be the set of A P SLp2, Rq k for which the semigroup xAy contains an elliptic element R (that is, such that |tr R| ă 2).The sets H and E are open and pairwise disjoint.We recall the following result: Therefore, to show that U :" H Y C is dense in SLp2, Rq k , we need to show: Lemma 6.4.C X E is dense in E.
Let I be the set of A P E such that xAy contains a matrix conjugate to an irrational rotation.Lemma 6.5.I is dense in E.
Proof.Let pA 1 , . . ., A k q P E, and fix an elliptic product A in . . .A i1 .Let P θ :" R θ A in . . .R θ A i1 , where R θ denotes the rotation by angle θ.By [ABY'10, Lemma A.4], the function θ Þ Ñ tr P θ has a nonzero derivative at θ " 0. Therefore we can find θ 0 arbitrarily close to 0 such that P θ0 is conjugate to an irrational rotation.Therefore pR θ0 A 1 , . . ., R θ0 A k q P I, proving the lemma.
Proof of Lemma 6.4.Let N be the set of A " pA 1 , . . ., A k q P SLp2, Rq k such that not all A i commute; then N is open and dense.We will show that N X I Ă C , (6.1) and so the desired result will follow from Lemma 6.5.Take A " pA 1 , . . ., A k q P N X I. Let R P xAy be conjugate to an irrational rotation.Since the sets N , I and C are invariant by conjugation, we can assume that R is an irrational rotation.Since A P N , there exists a generator A i that does not commute with R. Using the singular value decomposition of A i , we see that there exist n, m ě 0 such that H :" R n A i R m is a hyperbolic matrix.Let s be the contracting eigendirection of H. Now, given any unit vector v and any ε ą 0, we can find j ě 0 such that the unit vector R j v is sufficiently close to s, and so there exists ℓ ě 0 such that }H ℓ R j v} ă ε.This shows that A P C, thus proving (6.1) and the lemma.
As explained before, Theorem 3 follows.
Comparing to the present paper, the proof of Theorem 2 in [BBD'14] uses similar but slightly simpler arguments to get zero exponents.It does not obtain bounded norms, however.The present construction, especially Lemma 6.1, is more related to strategy suggested on [BBD'14,Remark 11.3].
Given a hyperbolic matrix L in SLp2, Rq, let u L , s L P P 1 denote its eigendirections, with u L corresponding to an eigenvalue of modulus bigger than 1.For convenience, the action of L on P 1 will also be denote by L.
Take A 1 , A 2 hyperbolic matrices in SLp2, Rq such that tr A 1 , tr A 2 ą 2 and tr A 1 A 2 ă ´2; then by [ABY'10,Prop. 3.4] there exists a cyclical order ă on P 1 such that Now take a hyperbolic matrix A 3 P SLp2, Rq such that (see Fig. 7): u A3 P ps A1 , u A1 q, s A3 P ps A2 , u A2 q, and A 3 u A2 " s A1 .
Fig. 7.The example of Proposition A.2.The thick part represents the "non-strict multicone" M .For each L, the arrow labelled L represents the hyperbolic geodesic from s L to u L .
Proposition A.2.The one-step cocycle generated by A :" pA 1 , A 2 , A 3 q has no λ 1 -minimizing measure.We note that the example is in the boundary of the hyperbolic component H Ă SLp2, Rq 3 described in [ABY'10,Prop. 4.16].
Before proving the proposition, let us describe a general geometrical construction.Consider a cocycle given by T : Ω Ñ Ω and A : Ω Ñ GLp2, Rq.Let S be the skew-product map on Ω ˆP1 induced by the cocycle.The derivative along the P 1 fiber of the map S at a point pω, xq P Ω ˆP1 is a linear map Lpω, xq : T x P 1 Ñ T Apωqx P 1 .
(A.3) Fix a rotation-invariant Riemannian metric on P 1 , and let f pω, xq denote the operator norm of Lpω, xq.Now suppose that µ is an ergodic T -invariant measure and μ is a S-invariant probability measure that projects to µ.Then we have the following fact (whose easy proof is left to the reader): Proof of Proposition A.2. Let A 1 , A 2 , A 3 be as above, and consider the one-step cocycle pT, Aq, where T is the shift on Ω :" t1, 2, 3u Z`, and A : Ω Ñ SLp2, Rq is given by Apωq " A ω0 .Let S be the induced skew-product map on Ω ˆP1 .Due to the "heteroclinic connection" A 3 u A2 " s A1 , the cocycle is not uniformly hyperbolic, and therefore λ ⊥ 1 pAq " 0. To prove the proposition we will show that λ 1 pA, µq ą 0 for every ergodic µ P M T .
Fix a point q in the interval pu A2A1 , s A2A1 q, and then a point p in the interval pA 1 q, A ´1 2 qq.Let M :" ps A1 , pq Y pu A2 , qq, as in Fig. 7. Then the set M is forwardinvariant under the projection action of each matrix A i .
Endow each connected component of M with its Riemannian Hilbert metric.Given a point pω, xq P Ω ˆM , let gpω, xq denote the operator norm of the linear map (A.3), where we take Hilbert metrics on both tangent spaces.Since the set ApωqpM q is contained in M and none of its connected components coincided with a connected component of M , we have gpω, xq ă 1.
Take any ergodic T -invariant measure µ, and lift it to a S-invariant measure μ supported on the forward S-invariant compact set Ω ˆM .We can assume that µ is neither δ 1 8 nor δ 2 8 , because otherwise λ 1 pA, µq ą 0 trivially.It is then easy to see that μ gives zero weight to the subset Ω ˆBM , and in particular the integral I :" ş log g dμ is well-defined.It is immediate from the definitions that log g ´log f is coboundary with respect to S, and therefore ş log f dμ " I. Since g ă 0, we have I ă 0 and so Lemma A.3 gives λ 1 pA, µq ‰ 0, as we wanted to show.
A.4. Examples of non-uniqueness of optimizing measures.Let us show that in the context of Theorem 2, the Mather sets K ⊤ and K ⊥ are not necessarily uniquely ergodic.In other words, the λ 1 -maximizing and λ 1 -minimizing measures can fail to be unique.
Take a pair of matrices A 1 and A 2 in GLp2, Rq 2 with respective eigenvalues χ 1 pA 1 q ą χ 2 pA 1 q and χ 1 pA 2 q ą χ 2 pA 2 q, all of them positive.Let v j pA i q P P 1 be the eigendirection of A i corresponding to the eigenvalue χ j pA i q.We can choose the pair A " pA 1 , A 2 q so that: ‚ the geodesics Ý ÝÝÝÝÝÝÝÝ Ñ v 2 pA 1 qv 1 pA 1 q and Ý ÝÝÝÝÝÝÝÝ Ñ v 2 pA 2 qv 1 pA 2 q cross; ‚ A has a forward invariant cone M Ă P 1 with the forward nonoverlapping property; ‚ A has a backwards invariant cone N Ă P 1 with the backards nonoverlapping property.
Proof of the claim.The four relations in (A.4) respectively imply: e 1 pξq P A 1 pM q, e 2 pξq P A ´1 2 pN q, e 1 pηq P A 2 pM q, e 2 pηq P A ´1 1 pN q.
Let 1 8 and 2 8 P t1, 2u Z be the two fixed points of the shift, and let ζ 12 and ζ 21 P k Z be the following "homoclinic points": It follows from Claim A.4 that K ⊤ is contained in the closure of the orbit of either ζ 12 or ζ 21 .Since K ⊤ equals the union of supports of the invariant probability measures that give full weight to K ⊤ itself, it follows that K ⊤ Ă t1 8 , 2 8 u.
Of course we can choose A 1 , A 2 such that additionally χ 1 pA 1 q " χ 1 pA 2 q; in this case K ⊤ equals t1 8 , 2 8 u and so it is not uniquely ergodic.
In a very similar way we produce an example where K ⊥ " t1 8 , 2 8 u.The only difference is that A " pA 1 , A 2 q are chosen so that the geodesics Ý ÝÝÝÝÝÝÝÝ Ñ v 2 pA 1 qv 1 pA 1 q and Ý ÝÝÝÝÝÝÝÝ Ñ v 2 pA 2 qv 1 pA 2 q are coparallel, and so if the points ξ, η satisfy (A.4) then the geodesics Ý ÝÝÝÝÝÝ Ñ e 2 pξqe 1 pξq and Ý ÝÝÝÝÝÝ Ñ e 2 pηqe 1 pηq cross.(See Fig. 9.) A.5. Open questions and directions for future research.There are several different directions along which one could try to extend the results of this paper.Notice that the NOC is indeed necessary for the validity of Theorem 2; an example is given in Remark 1.1 for α " β.However all the examples we know are very non-generic.So we ask whether the NOC can be replaced by a weaker condition, preferably one that is "typical" (open and dense) among k-tuples of matrices that generate dominated cocycles.
Regarding more general cocycles, we remark that there is also a notion of multicones for one-step cocycles over subshifts of finite type: see [ABY'10].It seems to be straightforward to adapt the arguments given here to that more general situation (and thus also for n-step cocycles) with appropriate nonoverlapping conditions, but we have not checked the details.
Even more generally, we would like to have results about Lyapunov-optimizing measures for cocycles that are not locally constant.We believe that some of the construction of this paper should extend to cocycles admitting unstable and stable holonomies (over a hyperbolic base dynamics).
Let us return to one-step cocycles over the full shift.A possible strengthening of the conclusions of Theorem 2 would be to replace subexponential complexity (zero entropy) by linear complexity (as in [BMa'02]), or polynomial complexity (as in [HMS'13]).Perhaps under generic conditions we can even obtain bounded complexity (periodic orbits), in the style of [Co].
Another line of study is to consider a relative Lyapunov-optimization problem for one-step cocycles where the frequencies of each matrix are fixed.The paper [JS'90] deals with a problem which can be reformulated in this terms.See [GL'07] for general results on relative optimization in the classical commutative setting.Let us also remark that this relative optimization setting is natural in the context of Lagrangian dynamics, where it corresponds to fixing the homology; see [Ma'91].
It should also be worthwhile to investigate the relations between Lyapunovoptimizing results as ours and the geometry of Riemann surfaces.
Regarding non-dominated one-step SLp2, Rq-cocycles, Theorem 3 says that we should not expect λ 1 -minimizing measures to have zero entropy.However, it seems likely that λ 1 -maximizing measures should have zero entropy.Notice that the corresponding Mather set (whose existence is given by [Mo'13]) is automatically uniformly hyperbolic.
Let us also remark that the only examples of k-tuples of matrices that do not satisfy the dichotomy of Theorem 3 (or Corollary A.1) are very particular ones (e.g., appropriate k-tuples with a common invariant direction).So we ask whether these counterexamples can be described explicitly, or at least whether they are contained in a finite union of submanifolds of positive codimension.
Of course most of the concepts and questions discussed in this paper make sense in higher dimension.In particular, we ask whether a higher-dimensional version of our zero entropy Theorem 2 (stated in terms of domination of index 1) holds true.As mentioned above, the construction of Barabanov functions can be adapted to this situation: see [BMo,§ 2.2].Lemma 4.2 should also be possible to extend: compare with [BMa'02, Prop.2.6].However, the rest of our proof relies on low-dimensional arguments.
Finally, we remark that the results obtained here can be considered as part of the multifractal analysis of Lyapunov exponents of linear cocycles, a broad field of study launched essentially by Feng [Fe'03].